arcsine distribution
Levy arcsine distribution
continuous
The arcsine distribution models the proportion of time that Brownian motion is positive
\( a \in (-\infty, \infty) \), location
\( w \in (0, \infty) \), scale
\( a = 0 \), \( w = 1 \)
\(x \in [a, a + w] \)
\(f(x) = \frac{1}{\pi\sqrt{(x - a) (a + w -x)}}\)
\(x \in \{a, a + w\} \)
\(F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x - a}{w}\right)\)
\(Q(u) = a + w \sin^2\left(\frac{\pi}{2} u\right), \; u \in (0, 1)\)
\( M(t) = e^{a t} \sum_{n=0}^\infty \left(\prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}\right) \frac{w^n t^n}{n!}, \; t \in (-\infty, \infty) \)
\(a + \frac{1}{2} w\)
\(\frac{1}{8} w^2\)
\(0\)
\(-\frac{3}{2}\)
\(a + \frac{1}{2} w\)
\(a + \frac{2 - \sqrt{2}}{4} w\)
\(a + \frac{2 + \sqrt{2}}{4} w\)
Derived by Paul Levy in 1939 as the distribution of proportion of time that Brownian motion is positive
arnold1980some
Bernoulli distribution
discrete
The Bernoulli distribution governs an indicator random variable
\(p \in [0, 1]\), the probability of the event
\( p = \frac{1}{2} \)
\(\{0, 1\}\)
\(f(x) = p^x (1 - p)^{1 - x}, \; x \in \{0, 1\}\)
\(\lfloor 2 p \rfloor\)
\(F(x) = (1 - p)^{1 - x}, \; x \in \{0, 1\}\)
\(Q(u) = F^{-1}(u), \; u \in [0, 1]\) where \(F\) is the distribution function
\(G(t) = 1 - p + p t, \; t \in (-\infty, \infty)\)
\(M(t) = 1 - p + p e^t, \; t \in (-\infty, \infty)\)
\(\varphi(t) = 1 - p + p e^{i t}, \; t \in (-\infty, \infty)\)
\(\mu(n) = p, \; n \in \{0, 1, \ldots\}\)
\(p\)
\(p (1-p)\)
\(\frac{1 - 2 p}{\sqrt{p (1 - p)}}\)
\(\frac{1- 6 p + 6 p^2}{p (1 - p)}\)
\(-(1 - p) \ln(1 - p) - p \ln(p)\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
power series
exponential
Named for Jacob Bernoulli
marshall1985family
beta distribution
continuous
The beta distribution is used to model random proportions and probabilities
\(\alpha \in (0, \infty)\), the left shape parameter
\(\beta \in (0, \infty)\), the right shape parameter
\( \alpha = 1 \), \( \beta = 1 \)
\((0, 1)\)
\(f(x) = \frac{1}{B(\alpha, \beta)} x^{\alpha-1}(1 - x)^{\beta-1}, \; x \in (0, 1)\), where \( B \) is the beta function
\(\frac{\alpha - 1}{\alpha + \beta - 2}; \; \alpha \in (1, \infty), \beta \in (1, \infty)\)
\(F(x) = \frac{B(x; \alpha, \beta)}{B(\alpha, \beta)}, \; x \in (0, 1)\), where \( x \mapsto B(x; \alpha, \beta) \) is the incomplete beta function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\), where \(F\) is the beta cumulative distribution function.
\(1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}\), for all real values \(t\)
\(\frac{\alpha}{\alpha + \beta}\)
\(\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\)
\(\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}\)
\(\frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)}\)
\(\ln(B(\alpha, \beta)) - (\alpha - 1) \psi(\alpha) - (\beta - 1) \psi(\beta) + (\alpha + \beta - 2) \psi(\alpha + \beta)\) where \(\psi\) is the digamma function
\(Q\left(\frac{1}{2}\right) \)
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
exponential
mcdonald1995generalization
beta general distribution
beta generalized distribution
generalized beta distribution
continuous
The generalized beta distribution is used to model random proportions and probabilities.
It extends the (standard) beta distribution, supported on \([0, 1]\) to an arbitrary range
\([L, R]\)
\(\alpha \in (0, \infty)\), the left shape parameter
\(\beta \in (0, \infty)\), the right shape parameter
\(L \in (-\infty, \infty)\), the left limit of the support range
\(R \in (L, \infty)\), the right limit of the support range
\((L, R)\)
R \\
\frac{\beta}{R - L} & \text{for } x == L \text{ and } \alpha == 1 \\
\infty & \text{for } x == L \text{ and } \alpha > 1 \\
0 & \text{for } x == L \text{ and } \alpha > 1 \\
\frac{\alpha}{R - L} & \text{for } x == R \text{ and } \beta == 1 \\
\infty & \text{for } x == R \text{ and } \beta < 1 \\
0 & \text{for } x == R \text{ and } \beta > 1 \\
\exp(\log\Gamma(\alpha + \beta) - (\log\Gamma(\alpha) + \log\Gamma(\beta)) - \log(R - L) +
(\alpha-1)*\log(\frac{x-L}{R-L})+(\beta-1)*\log(\frac{R-x}{R-L})) & otherwise
\end{cases}\)
]]>
\(I_{\frac{x-L}{R}}(\alpha,\beta)\!\)
\( \frac{\alpha * (R - L)}{\alpha + \beta}\)
\( \frac{(R-L)*(R-L)*\alpha*\beta}{(\alpha+\beta)^2*(\alpha+\beta+1)} \)
exponential
mcdonald1995generalization
inverse beta distribution
continuous
The inverse beta distribution, as the name suggests, is the inverse probability dsitribution of a
Beta-distributed variable
\(\alpha \in (0, \infty)\), the left shape parameter
\(\beta \in (0, \infty)\), the right shape parameter
\((0, 1)\)
\(f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\), \(B\) is the beta function
\(\frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\)
\(F(x) = B(\frac{x}{1+x}; \alpha,\beta) \), where \(B(x; \alpha,\beta)\) is the incomplete beta function
exponential
mcdonald1995generalization
binomial distribution
discrete
The binomial distribution models the number of successes in a fixed number of independent trials each with the same probability of success
\(n \in \{1, 2, \ldots\}\), the number of trials
\(p \in [0, 1]\), the probability of success
\( n = 1, \; p = \frac{1}{2} \)
\(\{0, 1, \ldots, n\}\)
\(f(x) = {n \choose x} p^x (1 - p)^{n - x}, \; x \in \{0, 1, \ldots, n\}\)
\(\lfloor (n + 1) p \rfloor\)
\(F(x) = B(1 - p; n - x, x + 1), \; x \in \{0, 1, \ldots, n\}\) where \(B\) is the incomplete beta function
\(Q(r) = F^{-1}(r), \; r \in [0, 1]\) where \(F\) is the distribution function
\(G(t) = (1 - p + p t)^n, \; t \in (-\infty, \infty)\)
\(M(t) = (1 - p + p e^t)^n, \; t \in (-\infty, \infty)\)
\(\varphi(t) = (1 - p + p e^{i t})^n, \; t \in (-\infty, \infty)\)
\(n p\)
\(n p (1 - p)\)
\(\frac{1 - 2 p}{\sqrt{n p (1 - p)}}\)
\(\frac{1 - 6 p (1 - p)}{n p (1 - p)}\)
\(\frac{1}{2} \log_2[2 \pi e n p (1 - p)] + O\left(\frac{1}{n}\right)\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{3}\right)\) where \(Q\) is the quantile function
power series
exponential
The binomial distribution is attributed to Jacob Bernoulli
altham1978two
beta-binomial distribution
discrete
The beta-binomial distribution arises when the success parameter in the binomial distribution is randomized and given a beta distribution
\(n \in \{1, 2, \ldots\}\), the number of trials
\(a \in (0, \infty)\), the left beta parameter
\(b \in (0, \infty)\), the right beta parameter
\( n = 1, \; a = 1, \; b = 1 \)
\(\{0, 1, \ldots, n\}\)
\( f(x) = \binom{n}{x} \frac{a^{[x]} b^{[n - x]}}{(a + b)^{[n]}} \), \( x \in \{0, 1, \ldots, n\} \) where \( m^{[j]} \) denotes the rising power of \( m \) of order \( j \)
\(F(x) = \sum_0^x f(t), \quad x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function
\(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function
\(_{2}F_{1}(-n, a; a + b; 1 - e^t) \)
\(\frac{n a}{a + b} \)
\(n \frac{n a b (a + b + n)}{(a + b)^2 (a + b + 1)}\)
\(\frac{(a + b + 2 n)(b - a)}{(a + b + 2)} \sqrt{\frac{1 + a + b}{n a b (n + a + b)}}\)
\(\frac{(a + b)^2 ( + 1 + b)}{n a b (a + b + 2)(a + b + 3)(a + b + n)} \left[(a + b)(a + b - 1 + 6 n) + 3 a b (n - 2) + 6 n^2 - \frac{3 a b n (6-n)}{a + b} - \frac{18 a b n^2}{(a + b)^2}\right]\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
altham1978two
beta-negative binomial distribution
discrete
The beta-negative binomial distribution arises when the success parameter in the negative binomial distribution is randomized and given a beta distribution
\(k \in \{1, 2, \ldots\}\), the number of trials
\(a \in (0, \infty)\), the left beta parameter
\(b \in (0, \infty)\), the right beta parameter
\( k = 1, \; a = 1, \; b = 1 \)
\(\{k, k + 1, \ldots \}\)
\( f(x) = \binom{n - 1}{x - 1} \frac{a^{[x]} b^{[n-x]}}{(a + b)^{[n]}} \), where \( r^{[j]} \) denotes the rising power of order \( j \)
\(F(x) = \sum_0^x f(t), \quad x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function
\(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function
\(k \frac{a + b - 1}{a - 1}\) if \( a \gt 1 \)
\( k \frac{a + b - 1}{(a - 1)(a - 2)}[b + k (a + b - 2)] - k^2 \left(\frac{a + b - 1}{a - 1}\right)^2 \) if \( a \gt 2 \)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
johnson2005univariate
Cauchy distribution
Cauchy-Lorentz distribution
Lorentz distribution
Breit-Wigner distribution
continuous
The general Cauchy distribution is the location-scale family associated with the standard Cauchy distribution
\(a \in (-\infty, \infty)\). the location parameter
\(b \in (0, \infty)\), the scale parameter
\(\displaystyle x \in (-\infty, \infty)\!\)
\( a = 0, \; b = 1 \)
\(\frac{1}{\pi b \, \left[1+\left(\frac{x - a}{b}\right)^2\right]}\!\)
\( a \)
\(\frac{1}{\pi}\arctan\left(\frac{x - a}{b} \right) + \frac{1}{2} \)
\(Q(p) = F^{-1}(p) = a + b \tan \left(\pi (p - \frac{1}{2}) \right), \; p \in (0, 1)\)
Does not exist
\(\varphi(t) = \exp(a i t - b |t|), \; t \in (-\infty, \infty)\)
Does not exist
Does not exist
Does not exist
Does not exist
\(\ln (4 \pi b ) \)
\( a \)
\(a - b\)
\(a + b\)
location
scale
The distribution was first used by Simeon Poisson in 1824 and was re-introduced by Augustin Cauchy in 1853. It is also named for Hendrick Lorentz
haas1970inferences
chi-square distribution
chi-squared distribution
continuous
The chi-square distribution governs the sum of squares of independent standard normal variable
\(n \in (0, \infty)\), degrees of freedom
\( n = 1 \)
\((0, \infty)\)
\(\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\; x^{\frac{n}{2}-1} e^{-\frac{x}{2}}\,\), where \(\Gamma\) is the Gamma function
\(n - 2, \; n \in [2, \infty)\)
\(\frac{1}{\Gamma\left(\frac{n}{2}\right)}\;\gamma\left(\frac{n}{2},\,\frac{x}{2}\right)\), where \(\gamma\) is the lower incomplete Gamma function
\(Q(p) = F^{-1}(p), \; p \in [0, 1)\) where \(F\) is the distribution function
\(M(t) = \frac{1}{(1 - 2 t)^{n/2}}, \; t \in (-\infty, \frac{1}{2})\)
\(\frac{1}{(1 - 2 i t^{n/2})}, \; t \in (-\infty, \infty)\)
\(n\)
\(2 n\)
\(\sqrt{8 / n}\,\)
\(12 / n\)
\(\frac{n}{2} + \ln[2 \Gamma(n / 2)] + (1 - n / 2) \psi(n / 2)\), where \(\psi\) is the Digamma function
\(\approx n \left(1 - \frac{2}{9n}\right)^3\)
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
exponential
The chi-square distribution was first used by Karl Pearson in 1900
lancaster2005chi
non-central chi-square distribution
non-central chi-squared distribution
continuous
The non-central chi-square distribution distribution is a generalization of the chi-squared distribution, which arises in the power analysis of statistical tests where the null distribution is asymptotically a chi-squared distribution; important examples of such tests are the likelihood ratio tests
\(k \in (0, \infty)\), degrees of freedom
\(\lambda \in (0, \infty)\), non-centrality parameter
\(x \in [0; +\infty)\,\)
\(\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2}
I_{k/2-1}(\sqrt{\lambda x})\)
\(F(x) = 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)\),
where \(Q_M(a,b)\) is the Marcum Q-function
\(\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}, \) for \( 2t \lt 1\)
\(\frac{1}{(1 - 2 i t^{n/2}}) \; t \in (-\infty, \infty)\)
\(k+\lambda\)
\(2(k+2\lambda)\)
\(\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}\)
\(\frac{12(k+4\lambda)}{(k+2\lambda)^2}\)
exponential
sankaran1959non
chi distribution
continuous
The chi distribution governs the square root of a variable with the chi-square distribution
\(n \in \{1, 2, \ldots\}\), the degrees of freedom
\(x\in[0;\infty)\)
\(\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}\)
\(\sqrt{k-1}\,\)for\(k\ge1\)
\(P(k/2,x^2/2)\,\)
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(\mu(k) = \frac{2^{k/2} \Gamma[(n+k)/2)]}{\Gamma(n/2)}, \; n \in \{1, 2, \ldots\}\) where \(\Gamma\) is the gamma function
\(\mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}\)
\(\sigma^2=k-\mu^2\,\)
\(\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)\)
\(\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)\)
\(\ln[\Gamma(n/2)] + \frac{1}{2} [n - \ln(2) - (n-1) \psi_0(n/2)]\) where \(\psi_0\) is the polygamma function
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
krishnaiah1963note
continuous uniform distribution
rectangular distribution
continuous
The continuous uniform distribution governs a point chosen at random from an interval. The continuous uniform distribution, aka rectangular distribution, is a family of probability distributions where all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. It is the maximum entropy probability distribution for a random variate X under no constraint other than that it is contained in the distribution's support
\(a \in (-\infty, \infty)\), location, the left endpoint
\( w \in (0, \infty) \), scale, the width of the interval
\(b = a + w\), the right endpoint
\( a = 0, \; w = 1 \)
\([a, b]\)
\(f(x) = \frac{1}{b - a}, \; x \in [a, b]\)
all \(x \in [a, b]\)
\(F(x) = \frac{x - a}{b - a}, \; x \in [a, b]\)
\(Q(p) = a + p (b - a). \; p \in [0, 1]\)
\(M(t) = \frac{e^{t b} - e^{t a}}{t (b - a)}, \; t \in (-\infty, \infty)\)
\(\varphi(t) = \frac{e^{i t b} - e^{i t a}}{i t (b - a)}, \; t \in (-\infty, \infty)\)
\(\mu(t) = \frac{b^{t+1} - a^{t+1}}{(t + 1)(b - a)}, \; t \in (0, \infty)\)
\(\frac{1}{2}(a + b)\)
\(\frac{1}{12} (b - a)^2\)
\(0\)
\(-\frac{6}{5}\)
\(\ln(b - a)\)
\(\frac{1}{2}(a + b)\)
\(\frac{3}{4} a + \frac{1}{4}b\)
\(\frac{1}{4} a + \frac{3}{4} b\)
location
scale
kuipers2006uniform
discrete uniform distribution
discrete
The discrete uniform distribution governs a point chosen at random from a discrete interval
\(a \in (-\infty, \infty)\), location, the left endpoint
\( h \in (0, \infty) \), scale, the step size
\( n \in \{1, 2, \ldots\} \), the number of points
\(b = a + (n - 1) h\), the right endpoint
\( a = 0, \; h = 1 \)
\(\{a, a + h, \ldots, a + (n - 1) h\}\)
\( f(x) = \frac{1}{n}, \; x \in \{a, a + h, \ldots, a + (n - 1) h\}\)
\(x \in \{a, a + h, \ldots, a + (n - 1) h\}\)
\(F(x) = \frac{1}{n}\left(\left \lfloor\ \frac{x - a}{h} + 1\right \rfloor \right), \; x \in [a, b] \)
\(Q(p) = a + \left(\lceil n p \rceil - 1\right) h, \; p \in (0, 1] \)
\(M(t) = \frac{1}{n} e^{t a} \frac{1 - e^{n t h}}{1 - e^{t h}}, \; t \in (-\infty, \infty)\)
\(\frac{1}{2}(a + b) = a + \frac{1}{2}(n - 1)h\)
\(\frac{1}{12}(n^2 - 1) h^2 = \frac{1}{12}(b - a)(b - a + h)\)
\(0\)
\(-\frac{6(n^2 + 1)}{5(n^2 - 1)}\,\)
\(\ln(n)\,\)
\(Q\left(\frac{1}{2}\right) \) where \( Q \) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \( Q \) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \( Q \) is the quantile function
location
scale
freund1967modern
exponential distribution
negative exponential distribution
continuous
The exponential distribution models the time between random points in the Poisson model
\(r \in (0, \infty)\), rate
\( b = 1 / r \), scale
\( r = 1 \)
\([0, \infty)\)
\(f(x) = r e^{-r x}, \; x \in [0, \infty)\)
\(0\)
\(F(x) = 1 - e^{-r x}, \; x \in [0, \infty)\)
\(Q(p) = \frac{- \ln(1 - p)}{r}, \; p \in [0, 1)\)
\(\left(1 - r t\right)^{-1}\,\)
\(\frac{r}{r - i t}, \; t \in (-\infty, \infty)\)
\(\frac{1}{r}\)
\(\frac{1}{r^2}\)
\(2\)
\(6\)
\(1 - \ln(r)\)
\(\frac{\ln(2)}{r}\)
\(\frac{\ln(4) - \ln(3)}{r}\)
\(\frac{\ln(3)}{r}\)
exponential
scale
The exponential distribution was named by Karl Pearson in 1895
siegrist2007exponential
exponential-logarithmic distribution
continuous
The exponential-logarithmic distribution models failure times of devices with decreasing failure rate
\(p \in (0, 1)\), the shape parameter
\(b \in (0, \infty)\), the scale parameter
\( p = \frac{1}{2}, \; b = 1 \)
\([0,\infty)\)
\(f(x) = \frac{1}{-\ln p} \frac{b(1 - p) e^{-x/b}}{1 - (1 - p) e^{-x/b}}, \; x \in [0, \infty)\)
\(0\)
\(F(x) = 1 - \frac{\ln(1 - (1 - p) e^{-x/b})}{\ln p}, \; x \in [0, \infty)\)
\(Q(u) = b \ln\left(\frac{1 - p}{1 - p^{1 - u}}\right), \; u \in (0, 1)\)
\(\mu(n) = -n! b^n \frac{L_{n+1}(1 - p)}{\ln(p)}, \; n \in \{0, 1, \ldots\}\) where \(L_{n+1}\) is the polylog function of order \(n + 1\)
\(-b \frac{L_2(1 - p)}{\ln(p)}\) where \(L_2\) is the polylog function of order \(2\).
\(-b^2 \frac{2 L_3(1 - p)}{ln(p)} - \frac{L_2^2(1 - p)}{b^2 \ln^2(p)}\) where \(L_n\) is the polylog function of order \(n\)
\(b \ln(1 + \sqrt{p})\)
\(b \ln\left(\frac{1 - p}{1 - p^{3/4}}\right)\)
\(b \ln\left(\frac{1 - p}{1 - p^{1/4}}\right)\)
scale
tahmasbi2008two
exponential power distribution
generalized error distribution
continuous
The exponential power distribution is a family of symmetric, unimodal distributions that generalizes the normal and Laplace families
\(\mu \in (-\infty, \infty)\), the location parameter
\(\alpha \in (0, \infty)\), the scale parameter
\(\beta \in (0, \infty)\), the shape parameter
\(x \in (-\infty; +\infty)\!\)
\(f(x) = \frac{\beta}{2 \alpha \Gamma(1/\beta)} \exp\left[-\left(\frac{|x - \mu|}{\alpha}\right)^\beta\right], \; x \in (-\infty, \infty)\) where \(\Gamma\) is the gamma function
\(\mu\)
\(F(x) = \frac{1}{2} + \frac{\sgn(x - \mu)}{2 \Gamma (1 / \beta)} \gamma\left[\frac{1}{\beta}, \left(\frac{|x - \mu|}{\alpha}\right)^\beta\right], \; x \in (-\infty, \infty)\), where \(\Gamma\) is the gamma function and \(\gamma\) is the lower incomplete gamma function
\(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function
\(\mu\)
\(\frac{\alpha^2 \Gamma(3/\beta)}{\Gamma(1/\beta)}\) where \(\Gamma\) is the gamma function
\(0\)
\(\frac{\Gamma(5/\beta) \Gamma(1/\beta)}{\Gamma^2(3/\beta)} - 3\) where \(\Gamma\) is the gamma function
\(\frac{1}{\beta} - \log\left[\frac{\beta}{2 \alpha \Gamma(1/\beta)}\right]\) where \(\Gamma\) is the gamma function
\(\mu\)
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
location
scale
zhu2009properties
F-distribution
Snedecor's F-distribution
Fisher-Snedecor distribution
continuous
The F-distribution governs the ratio of independent, scaled chi-square variables
\(n \in (0, \infty)\), numerator degrees of freedom
\(d \in (0, \infty)\), denominator degrees of freedom
\( n = 1, \; d = 1 \)
\([0, \infty)\)
\( f(x) = \frac{\Gamma[(n + d)/2]}{\Gamma(n/2) \Gamma(d/2)} \left(\frac{n}{d}\right)^{n/2} \frac{x^{(n-2)/2}}{[1 + (n/d) x]^{(n+d)/2}}, \; x \in [0, \infty) \)
\(\frac{n - 2}{n} \frac{d}{d + 2} \) for \(n \gt 2\)
\(F(x) = \frac{B[n x / (n x + d), n/2, d/2]}{B(n/2, d/2)}, \; x \in [0, \infty)\), where \( B \) is the beta function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
Does not exist
\(\frac{d}{d - 2}\) for \(d \gt 2\)
\(\frac{2 d (n + d - 2)}{n (d - 2)^2 (d - 4)}\) for \(d \gt 4\)
\(\frac{(2 n + d - 2) \sqrt{8(d - 4)}}{(d - 6)\sqrt{n (n + d - 2)}}\) for \(d \gt 6 \)
\(\frac{20 d - 8 d^2 + d^3 + 44 n - 32 n d + 5 n d^2 - 22 n^2 - 5 n^2 d - 16}{n (d - 6)(d - 8)(n + d - 2)/12}\) for \( d \gt 8 \)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
The \(F\)-distribution was first derived by George Snedecor in 1934. The letter F was chosen as a tribute to Ronald Fisher
johnson2005univariate
gamma distribution
continuous
The gamma distribution governs the arrival times in the Poisson model, and has many applications in statistics
\(k \in (0, \infty)\), the shape parameter
\(b \in (0, \infty)\), the scale parameter
\( k = 1, \; b = 1 \)
\((0,\,\infty) \)
\(f(x) = \frac{1}{\Gamma(k) b^k} x^{k - 1}e^{-x / b}, \; x \in (0, \infty) \) where \( \Gamma \) is the gamma function
\((k - 1) b \) for \( k \gt 1 \)
\( F(x) = \frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{b}\right) \) where \( \gamma \) is the incomplete gamma function
\(Q(p) = F^{-1}(p)\) where \(F\) is the distribution function
\(M(t) = \frac{1}{(1 - b t)^k}, \; t \in (-\infty, 1 / b)\)
\(\varphi(t) = \frac{1}{(1 - i b t)^k}, \; t \in (-\infty, \infty)\)
\( k b\)
\( k b^2 \)
\( \frac{2}{\sqrt{k}} \)
\( \frac{6}{k} \)
\(\ln (4 \pi b)\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
scale
exponential
siegrist2007exponential
geometric distribution
discrete
The geometric distribution models the trial number of the first success in a sequence of Bernoulli trials
\(p \in (0, 1]\), the success parameter
\( p = \frac{1}{2} \)
\(\{1, 2, 3, \ldots\}\)
\(f(k) = p (1 - p)^{k - 1}, \; k \in \{1, 2, \ldots\}\)
\(1\)
\( F(k) = 1 - (1 - p)^k \)
\(Q(u) = \left\lceil \frac{\ln(1 - u)}{\ln(1 - p)} \right\rceil, \; u \in [0, 1)\)
\(P(t) = \frac{p t}{1 - (1 - p)t}, \; t \in \left(-\frac{1}{1 - p}, \frac{1}{1 - p}\right)\)
\(M(t) = \frac{p e^t}{1 - (1 - p) e^t}, \; t \in (-\infty, -\ln(1 - p))\)
\(\varphi(t) = \frac{p e^{i t}}{1 - (1 - p) e^{i t}}, \; t \in (-\infty, \infty)\)
\(\frac{1}{p}\)
\(\frac{1 - p}{p^2}\)
\(\frac{2 - p}{\sqrt{1 - p}}\!\)
\(6 + \frac{p^2}{1 - p}\)
\(\frac{-(1 - p) \log_2 (1 - p) - p \log_2 p}{p} \)
\(\left\lceil \frac{-\ln(2)}{\ln(1 - p)} \right\rceil\)
\(\left\lceil \frac{\ln(3) - \ln(4)}{\ln(1 - p)} \right\rceil\)
\(\left\lceil \frac{-\ln(4)}{\ln(1 - p)} \right\rceil\)
power series
exponential
The geometric distribution was used very early in the history of probability, but the name has been attributed to William Feller in 1950
philippou1983generalized
hypergeometric distribution
discrete
The hypergeometric distribution governs the number of objects of a given type when sampling without replacement from a multi-type population
\(N\), the population size
\(m\), the number of type 1 objects in the population
\(n\), the sample size
\( \left\{\max{(0,\, n+m-N)},\, \dots,\, \min{(m,\, n )}\right\}\)
\( f(x) = \frac{\binom{m}{x} \binom{N - m}{n - x}}{\binom{N}{n}}, \; x \in \left\{\max{(0,\, n+m-N)},\, \dots,\, \min{(m,\, n )}\right\} \)
\(\left \lfloor \frac{(n + 1)(m + 1)}{N + 2} \right \rfloor\)
\(1-{{{n \choose {k+1}}{{N-n} \choose {m-k-1}}}\over {N \choose m}} \,_3F_2\!\!\left[\begin{array}{c}1,\ k+1-m,\ k+1-n \\ k+2,\ N+k+2-m-n\end{array};1\right]\),
where \(_pF_q(a,b;c;z)\) is the generalized hypergeometric function.
\(\frac{{N-m \choose n} \scriptstyle{\,_2F_1(-n, -m; N - m - n + 1; e^{t}) } }
{{N \choose n}} \,\!\)
\(n \frac{m}{N}\)
\( n \frac{m}{N} \frac{N - m}{N} \frac{N - n}{N - 1} \)
\(\frac{(N - 2 m)\sqrt{N - 1}(N - 2 n)}{\sqrt{n m(N - m)(N - n)}(N - 2)}\)
\(\left[ \frac{N^2 (N-1)}{n(N - 2)(N - 3)(N - n)}\right] \left[ \frac{N(N+1) - 6 N(N - n)}{m (N - m)} + \frac{3 n (N - n)(N + 6)}{N^2} - 6 \right]\)
The hypergeometric distribution is very old, and was used by Jacob Bernoulli, Abraham DeMoivre, and others. The named was coined by H.T. Gonin in 1936
harkness1965properties
hyperbolic secant distribution
continuous
The hyperbolic secant distribution is a symmetric, unimodal distribution but with larger kurtosis than the normal distribution
\( \mu \in (-\infty, \infty) \), the location parameter
\( \sigma \in (0, \infty) \), the scale parameter
\( \mu = 0, \; \sigma = 1 \)
\((-\infty, \infty)\)
\( f(x) = \frac{1}{2 \sigma} \sech\left[\frac{\pi}{2}\left(\frac{x - \mu}{\sigma}\right)\right], \; x \in (-\infty, \infty) \)
\(\mu\)
\( F(x) = \frac{2}{\pi} \arctan\left\{\exp\left[\frac{\pi}{2} \left(\frac{x - \mu}{\sigma} \right) \right]\right\}, \; x \in (-\infty, \infty) \)
\(Q(p) = \mu + \sigma \frac{2}{\pi} \ln[\tan(\frac{\pi}{2} p)], \; p \in (0, 1)\)
\(M(t) = e^{\mu t} \sec(\sigma t), \; t \in (-\frac{\pi}{1}, \frac{\pi}{2 \sigma})\)
\(\mu\)
\(\sigma^2\)
\(0\)
\(2\)
\(\mu\)
\( \mu + \sigma \frac{2}{\pi} \ln(\sqrt{2} - 1)\)
\( \mu + \sigma \frac{2}{\pi} \ln(\sqrt{2} + 1)\)
location
scale
harkness1968generalized
Irwin-Hall distribution
continuous
The Irwin-Hall distribution governs the sum of \(n\) independent variables, each uniformly distributed on \([0, 1]\)
\(n \in \{1, 2, \ldots\}\), the number of terms
\( n = 1 \)
\([0, n]\)
\(f(x) = \frac{1}{2 (n - 1)!} \sum_{k=0}^n (-1)^k \binom{n}{k}\sgn(x - k)(x - k)^{n-1}, \; x \in [0, n]\)
\( n/2 \) for \( n \ge 2 \)
\( F(x) = \frac{1}{2} + \frac{1}{2 n!} \sum_{k=0}^n (-1)^k \binom{n}{k} \sgn(x - k) (x - k)^n, \; x \in [0, n]\)
\(M(t) = \left(\frac{e^t - 1}{t}\right)^n, \; t \in (-\infty, \infty)\)
\(\frac{n}{2}\)
\(\frac{n}{12}\)
\(\frac{n}{2}\)
The Irwin-Hall distribution is named for Joseph Irwin and Phillip Hall who independently analyzed the distribution in 1927
hall1927distribution
inverted beta distribution
beta prime distribution
beta distribution of the second kind
continuous
The inverted beta distribution is conjugate for the odds in the Bernoulli distribution
\(\alpha \in (0, \infty)\), the first shape parameter
\(\beta \in (0, \infty)\), the second shape parameter
\( \alpha = 1, \; \beta = 1 \)
\(x > 0\!\)
\(f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!\), \(B\) is the beta function
\(\frac{\alpha - 1}{\beta + 1}\) if \(\alpha \in [1, \infty)\)
\(F(x) = \int_0^x f(t) dt, \; x \in (0, \infty)\), where \(f\) is the probability density function.
Alternatively, \(F(x)=B(\frac{x}{1+x}; \alpha,\beta) \), where \( B(x; \alpha,\beta)\) is the incomplete beta function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(\frac{\alpha}{\beta - 1}\) if \(\beta \in (1, \infty)\)
\(\frac{\alpha (\alpha + \beta - 1)}{(\beta - 2)(\beta - 1)^2}\) if \(\beta \in (2, \infty)\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
mcdonald1995generalization
Laplace distribution
double exponential distribution
continuous
The Laplace distribution is a symmetric, unimodal distribution with tails that are fatter than those of the normal distribution
\(a \in (-\infty, \infty)\), location
\(b \in (0, \infty)\), scale
\( a = 0, \; b = 1 \)
\( (-\infty, \infty) \)
\(f(x) = \frac{1}{2 b} \exp \left(-\frac{\left|x - a\right|}{b} \right), \; x \in (-\infty, \infty) \)
\( a \)
\( F(x) = \begin{cases} \frac{1}{2} \exp\left(\frac{x - a}{b}\right), & x \in (-\infty, a] \\ 1 - \frac{1}{2} \exp\left(-\frac{x - a}{b}\right), & x \in [a, \infty) \end{cases} \)
\(Q(p) = a + b \ln(2 \min\{p, 1 - p\}), \; p \in (0, 1)\)
\(M(t) = \frac{e^{a t}}{1 - b^2 t}, \; t \in (-\frac{1}{b}, \frac{1}{b})\)
\(\varphi(t) = \frac{e^{a i t}}{1 + b^2 t}, \; t \in (-\infty, \infty)\)
\( a \)
\(2 b^2\)
\( 0 \)
\( 3 \)
\(\log(2 e b)\)
\( a \)
\(a - b \ln(2)\)
\(a + b \ln(2)\)
location
The Laplace distribution is named for Pierre Simon Laplace
kotz2001laplace
Levy distribution
van der Waals profile
stable distribution
continuous
The Levy distribution is a stable distribution that has applications in spectroscopy
\(a \in (-\infty, \infty)\), the location parameter
\(b \in (0, \infty)\), the scale parameter
\( a = 0, \; b = 1 \)
\((a, \infty)\)
\( f(x) = \sqrt{\frac{b}{2 \pi}} \frac{1}{(x - a)^{3/2}} \exp\left[-\frac{b}{2 (x - a)}\right], \; x \in (a, \infty)\)
\( a + \frac{1}{3} b\)
\( F(x) = 2 \left[1 - \Phi\left(\sqrt{\frac{b}{x - a}}\right)\right], \; x \in (a, \infty) \) where \( \Phi \) is the standard normal distribution function
\( F^{-1}(p) = a + \frac{b}{\left[\Phi^{-1}(1 - p / 2)\right]^2}, \; p \in [0, 1) \) where \( \Phi^{-1} \) is the standard normal quantile function
\(\varphi(t) = \exp\left(i a t - \sqrt{-2 i b t}\right), \; t \in (-\infty, \infty)\)
\(\infty\)
\(\infty\)
undefined
undefined
\(\frac{1}{2}[1 + 3 \gamma + \ln(16 \pi b^2)]\) where \(\gamma\) is Euler's constant
\( a + b \left[\Phi^{-1}\left(\frac{3}{4}\right)\right]^{-2}\) where \(\Phi^{-1}\) is the standard normal quantile function
\( a + b \left[\Phi^{-1}\left(\frac{7}{8}\right)\right]^{-2}\) where \(\Phi^{-1}\) is the standard normal quantile function
\( a + b \left[\Phi^{-1}\left(\frac{5}{8}\right)\right]^{-2}\) where \(\Phi^{-1}\) is the standard normal quantile function
location
scale
stable
The Levy distribution is named for Paul Pierre Levy
barndorff2001levy
Landau distribution
continuous
The Landau distribution is used in physics to describe the fluctuations in the energy
loss of a charged particle passing through a thin layer of matter. This distribution is a
special case of the stable Levy distribution with parameters (1, 1)
\(\mu \in (-\infty, \infty)\), the location parameter
\(c \in (0, \infty)\), the scale parameter
\(1, \infty)\)
\(f(x) = \sqrt{\frac{1}{2 \pi}} \frac{e^{-1/2(x - 1)}}{(x - 1)^{3/2}}, \;
x \in (1, \infty)\)
\(1 + \frac{1}{3}\)
\(F(x) = \int_1^x f(t) dt, \; x \in \) where \(f\) is the probability density function
landau1944energy
logarithmic distribution
logarithmic series distribution
log-series distribution
discrete
The logarithmic distribution is sometimes used to model relative species abundance
\(p \in (0, 1)\), the shape parameter
\( p = \frac{1}{2} \)
\(\{1, 2, 3, \ldots\}\)
\(f(k) = \frac{-1}{\ln(1 - p)} \frac{p^k}{k}, \; k \in \{1, 2, \ldots\}\)
\(1\)
\(F(k) = 1 + \frac{B(p; k + 1, 0)}{\ln(1 - p)}, \; k \in \{1, 2, \ldots\} \) where \( B \) is the incomplete beta function
\(G(t) = \frac{\ln(1 - p t)}{\ln(1 - p)}, \; t \in (-\frac{1}{p}, \frac{1}{p})\)
\(M(t) = \frac{\ln(1 - p e^t)}{\ln(1 - p)}, \; t \in (-\infty, -\ln(p))\)
\(\varphi(t) = \frac{\ln(1 - p e^{i t})}{\ln(1 - p)}, \; t \in (-\infty, \infty)\)
\(\frac{-1}{\ln(1 - p)} \frac{p}{1 - p}\!\)
\(-p \frac{p + \ln(1 - p)}{(1 - p)^2 \ln^2(1 - p)}\!\)
power series
The logarithmic distribution was first derived by Ronald Fisher in 1943
fisher1943relation
logistic distribution
continuous
The logistic distribution occurs in logistic regression
\(a \in (-\infty, \infty)\), the location parameter
\(b \in (0, \infty)\), the scale parameter
\( a = 0, \; b = 1 \)
\((-\infty, \infty)\)
\(f(x) = \frac{e^{-(x - a)/b}}{b \left(1 + e^{-(x - a)/b}\right)^2}, \; x \in (-\infty, \infty)\)
\( a \)
\(F(x) = \frac{1}{1 + e^{-(x - a)/b}}, \; x \in (-\infty, \infty)\)
\(Q(p) = a + b \ln\left(\frac{p}{1 - p}\right), \; p \in (0, 1)\)
\(M(t) = e^{a t} B(1 - b t, 1 + b t)\) where \(B\) is the beta function
\( a \)
\( \frac{\pi^2}{3} b^2 \)
\( 0 \)
\( 6/5 \)
\(\ln(b) + 2\)
\( a \)
\(a - \ln(3) b\)
\(a + \ln(3) b\)
location
scale
Logistic regression was first used by D.R. Cox in 1958
balakrishnan1992handbook
generalized logistic distribution
skew logistic distribution
continuous
The generalized logistic distribution represents several different families of
probability distributions. One family is called the skew-logistic distribution.
Other families of distributions that have also been called generalized
ogistic distributions include the shifted log-logistic distribution,
which is a generalization of the log-logistic distribution
\(\alpha >0\), the location parameter
\(\beta >0\), the scale parameter
\((-\infty, \infty)\)
\(f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\frac{\exp(-\beta x)}
{(1+\exp(-x))^{\alpha+\beta}}\), where \(B\) is the beta function.
balakrishnan2009continuous
log-normal distribution
log normal distribution
lognormal distribution
Galton distribution
continuous
The log-normal distribution models certain skewed variables
\(\mu \in (-\infty, \infty)\), the normal mean
\(\sigma \in (0, \infty)\), the normal standard deviation
\( \mu = 0, \; \sigma = 1 \)
\((0, \infty)\)
\(f(x) = \frac{1}{\sigma x \sqrt{2 \pi}}\, \exp\left(-\frac{\left[\ln(x) - \mu\right]^2}{2 \sigma^2}\right), \; x \in (0, \infty) \)
\(e^{\mu - \sigma^2}\)
\( F(x) = \Phi\left[\frac{\ln(x) - \mu}{\sigma}\right], \; x \in (0, \infty)\) where \( \Phi \) is the standard normal distribution function
\(F^{-1}(p) = \exp\left[\mu + \sigma \Phi^{-1}(p)\right]\), where \(\Phi\) is the standard normal distribution function
\(\mu(n) = \exp(\mu n + \frac{1}{2} \sigma^2 n^2), \; n \in \{0, 1, \ldots\}\)
\(e^{\mu + \sigma^2/2}\)
\( e^\mu \)
\((e^{\sigma^2} - 1) e^{2 \mu + \sigma^2}\)
\((e^{\sigma^2} + 2) \sqrt{e^{\sigma^2} - 1}\)
\(e^{4 \sigma^2} + 2 e^{3 \sigma^2} + 3e^{2 \sigma^2} - 6\)
\(\frac12 + \frac12 \ln(2 \pi \sigma^2) + \mu\)
\(e^{\mu}\,\)
\(\exp\left[\mu + \sigma \Phi^{-1}\left(\frac{1}{4}\right)\right]\), where \(\Phi\) is the standard normal distribution function
\(\exp\left[\mu + \sigma \Phi^{-1}\left(\frac{3}{4}\right)\right]\), where \(\Phi\) is the standard normal distribution function
scale
exponential
The lognormal distribution was first studied by Donald McAlister in 1879, in response to a problem posed by Francis Galton. This historical origin is the reason for the alternative name Galton distribution. The term lognormal distribution was first used by J.H. Gaddum in 1945
famoye1995continuous
log-logistic distribution
Fisk distribution
continuous
The log-logistic distribution models lifetimes of devices whose failure rates at first increase and then decrease
\(k \in (0, \infty)\), the shape parameter
\(b \in (0, \infty)\), the scale parameter
\([0, \infty)\)
\( k = 1, \; b = 1 \)
\( f(x) = \frac{b^k k x^{k - 1}}{(b^k + x^k)^2}, \; x \in [0, \infty)\)
\( b \left(\frac{k - 1}{k + 1}\right)^{1/k} \) if \( a \gt 1 \)
\(F(x) = \frac{x^k}{b^k + x^k}, \; x \in [0, \infty)\)
\(F^{-1}(p) = b \left(\frac{p}{1 - p}\right)^{1/k}, \; p \in [0, 1)\)
\(\mu(n) = b^n \frac{\pi n / k}{\sin(\pi n / k)}, \; n \lt k\)
\( \infty \) if \( 0 \lt k \le 1 \); \(b \frac{\pi / k}{\sin(\pi / k)}\) if \( k \gt 1 \)
does not exist if \( 0 \lt k \le 1 \); \( \infty \) if \( 1 \lt k \le 2 \); \( b^2 \left[\frac{2 \pi / k}{\sin(2 \pi / k)} - \frac{\pi^2 / k^2}{\sin^2(\pi / k)}\right] \) if \( k \gt 2 \)
\( b\)
\( b (1/3)^{1/k} \)
\(b 3^{1/k} \)
scale
The log-logistic distribution is known as the Fisk distribution by economists. P.R. Fisk used the distribution to model income in 1961
shoukri1988sampling
Maxwell-Boltzmann distribution
Maxwell distribution
continuous
The Maxwell-Boltzmann Distribution arises in the kinetic theory of gases
\(b \in (0, \infty)\), the scale parameter
\( b = 1 \)
\([0, \infty)\)
\(f(x) = \frac{1}{b^3} \sqrt{\frac{2}{\pi}} x^2 \exp\left(-\frac{x^2}{2 b^2}\right), \; x \in [0, \infty)\)
\(\sqrt{2} b\)
\(F(x) = 2 \Phi\left(\frac{x}{b}\right) - \frac{1}{b} \sqrt{\frac{2}{\pi}} x \exp\left(-\frac{x^2}{2 b^2}\right) - 1\) where \( \Phi \) is the standard normal distribution function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(2 b \sqrt{\frac{2}{\pi}}\)
\(\frac{b^2 (3 \pi - 8)}{\pi}\)
\(\frac{2 \sqrt{2}(16 - 5 \pi)}{(3 \pi - 8)^{3/2}}\)
\(4 \frac{(-96 + 40 \pi - 3 \pi^2)}{(3 \pi - 8)^2}\)
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
scale
The Maxwell-Boltzman distribution is named for James Clerk Maxwell and Ludwig Boltzmann for their use of the distribution is modeling the energy of molecules in a gas
laurendeau2005statistical
negative binomial distribution
Pascal distribution
discrete
The negative binomial distribution governs the number of trials (x) needed for a specified number of successes (k) in the Bernoulli trials model
\(k \in \{1, 2, \ldots\}\), the number of successes
\(p \in (0, 1]\), the success parameter
\( k = 1, \; p = \frac{1}{2} \)
\(\{k, k+1, \ldots\}\)
\(f(x) = \binom{x - 1}{k - 1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}\)
\(\lfloor 1 + \frac{k-1}{p}\rfloor\)
\(F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}\) where \(f\) is the probability density function. It can
also be expressed as \(1-I_p(x-r+1,\,k) \), where \(I_p\) is the regularized incomplete beta function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(G(t) = \left[\frac{p t}{1 - (1-p) t}\right]^k, \; t \in (-\frac{1}{1-p}, \frac{1}{1-p})\)
\(M(t) = \left[\frac{p e^t}{1 - (1-p) e^t}\right]^k, \; t \in (-\infty, -\ln(1 - p))\)
\(\varphi(t) = \left[\frac{p e^{i t}}{1 - (1-p) e^{i t}}\right]^k, \; t\in (-\infty, \infty)\)
\(k \frac{1}{p}\)
\(k \frac{1-p}{p^2}\)
\(\frac{2-p}{\sqrt{k (1-p)}}\)
\(\frac{1}{k} \left[6 + \frac{p^2}{1 - p}\right]\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
The alternative name Pascal distribution is in honor of Blaise Pascal who used the distribution in his solution to the Problem of Points
el2006negative
normal distribution
Gaussian distribution
error distribution
continuous
The normal distribution is used to model physical quantities that are subject to numerous small, random errors
\(\mu \in (-\infty, \infty)\), the location parameter
\(\sigma \in (0, \infty)\), the scale parameter
\( \mu = 0, \; \sigma = 1 \)
\((-\infty, \infty)\)
\(f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2 \right], \; x \in (-\infty, \infty)\)
\(\mu\)
\(F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right), \; x \in (-\infty, \infty)\) where \(\Phi\) is the standard normal distribution function
\(Q(p) = \mu + \sigma \Phi^{-1}(p), \; p \in (0, 1)\)
\(M(t) = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right), \; t \in (-\infty, \infty)\)
\(\varphi(t) = \exp\left(i \mu t - \frac{1}{2} \sigma^2 t^2\right), \; t \in (-\infty, \infty)\)
\(\mu\)
\(\sigma^2\)
\(0\)
\(0\)
\(\frac{1}{2} \ln(2 \pi e \sigma^2)\)
\(\mu\)
\(\mu - \Phi^{-1}\left(\frac{1}{4}\right) \sigma\)
\(\mu + \Phi^{-1}\left(\frac{1}{4}\right) \sigma\)
location
scale
exponential
stable
dinov2008central
The normal distribution was first derived by Carl Friedrich Gauss in 1809 (hence the alternative name Gaussian distribution). The normalizing constant and the first version of the Central Limit Theorem were contributions by Pierre Simon Laplace. The term normalizing constant was popularized by Karl Pearson around the turn of the 20th century
students t distribution
Student's t distribution
t distribution
continuous
The t distribution arises when estimating the mean of a normally distributed population when the sample size is small and population standard deviation is unknown. It comes into play in various statistical analyses like Students t-test for assessing the between-group statistical significant differences of two sample means, construction of confidence intervals for difference between two population means, linear regression analyses, etc. Like the normal distribution, the t-distribution is symmetric, bell-shaped and unimodal. However it has heavier tails, meaning that it is more prone to producing values that fall far from its mean. The Students t-distribution is a special case of the generalized hyperbolic distribution
\(n \in (0, \infty)\), degrees of freedom
\( n = 1 \)
\((-\infty, \infty)\)
\(f(x) = \frac{\Gamma \left(\frac{n + 1}{2} \right)} {\sqrt{n \pi} \Gamma \left(\frac{n}{2} \right)} \left(1 + \frac{x^2}{n} \right)^{-\frac{n + 1}{2}}, \; x \in (-\infty, \infty)\) where \( \Gamma \) is the gamma function
\(0\)
\(F(x) = \frac{1}{2} + x \Gamma \left( \frac{n + 1}{2} \right) \frac{\,_2F_1 \left ( \frac{1}{2},\frac{n + 1}{2};\frac{3}{2}; - \frac{x^2}{n} \right)} {\sqrt{\pi n}\,\Gamma \left(\frac{n}{2}\right)}, \; x \in (-\infty, \infty)\), where \({ }_2F_1\) is the hypergeometric function
0 for \( n \gt 1 \)
\(\frac{n}{n - 2}, \) for \( n \gt 2 \)
0 for \( n \gt 3 \)
\(\frac{6}{n - 4}, \) for \( n \gt 4\)
\(\frac{n + 1}{2}\left[\psi \left(\frac{1 + n}{2} \right) - \psi \left(\frac{n}{2} \right) \right] + \log{\left[\sqrt{n}
B \left(\frac{n}{2}, \frac{1}{2} \right)\right]} \), where \(\psi\) is the digamma function and \(B\) is the beta function
0
exponential
li1957student
truncated normal distribution
continuous
The truncated normal distribution is the probability distribution of a normally
distributed random variable (\(X \sim N(\mu, \sigma^2) \)) whose value is either bounded below, above or on both sides.
The truncated normal distribution has wide applications in statistics and econometrics
\(\mu \in (-\infty, \infty)\), the location parameter
\(\sigma \in (0, \infty)\), the scale parameter
\(a \in (-\infty, \infty)\), left limit
\(b \in (a, \infty)\), right limit ( \(a \lt b\) )
\([a, b]\)
\(f(x;\mu,\sigma,a,b)=\frac{1}{\sigma Z}\phi(\xi)\)
b\end{array}\right.\)
]]>
\(F(x;\mu,\sigma,a,b)=\frac{\Phi(\xi)-\Phi(\alpha)}{Z}\), where
\( \phi(\xi) = \frac{1}{\sqrt{2 \pi}} \exp{(-\frac{1}{2}\xi^2)} \) is the probability density function of
the standard normal distribution, \( \Phi(\cdot) \) is the standard normal
cumulative distribution function, \( Z=\Phi(\beta)-\Phi(\alpha) \),
\( \xi=\frac{x-\mu}{\sigma}\), \(\alpha=\frac{a-\mu}{\sigma}\), \(\beta=\frac{b-\mu}{\sigma}\).
\(\mu+\frac{\phi(\alpha)-\phi(\beta)}{Z}\sigma\)
\(\sigma^2\left[1+\frac{\alpha\phi(\alpha)-\beta\phi(\beta)}{Z}-\left(\frac{\phi(\alpha)-\phi(\beta)}{Z}\right)^2\right]\)
kotz2000continuous
Pareto distribution
Bradford distribution
continuous
The Pareto distribution models highly skewed variables that sometimes arise in economics
\(a \in (0, \infty)\), the shape parameter
\(b \in (0, \infty)\), the scale parameter
\( a = 1, \; b = 1 \)
\([b, \infty)\)
\(f(x) = \frac{a b^a}{x^{a+1}}, \; x \in [b, \infty)\)
\(b\)
\(F(x) = 1 - \left(\frac{b}{x}\right)^a, \; x \in [b, \infty)\)
\(F^{-1}(p) = \frac{b}{(1 - p)^{1/a}}, \; p \in [0, 1)\)
\(\varphi(t) = a (-i b t)^a \gamma(-a, -i b t)\) where \(\gamma\) is the lower incomplete gamma function
\(\mu(n) = b^n \frac{a}{a - n}, \; n \in (0, k)\)
\( b \frac{a}{a - 1} \) for \( a \gt 1 \)
\( b^2 \frac{a}{(a - 1)^2 (a - 2)} \) for \( a \gt 2 \)
\(\frac{2(1 + a)}{a - 3} \sqrt{\frac{a - 2}{a}}\) for \(a \gt 3\)
\(\frac{6(a^3 + a^2 - 6 a - 2)}{a(a - 3)(a - 4)}\) for \(a \gt 4 s\)
\(\ln\left(\frac{b}{a}\right) + \frac{1}{a} + 1\)
\(b 2^{1/a}\)
\(b \left(\frac{4}{3}\right)^{1/a}\)
\(b 4 ^{1/a}\)
scale
The Pareto distributin is named for the Italian economist Vilfredo Pareto, who used the distribution to model wealth, income and other economic variables.
arnold1985pareto
Poisson distribution
discrete
The Poisson distribution models the number of random points in a region of time or space under certain ideal conditions
\(\lambda \in (0, \infty)\), the (shape, mean, or rate) parameter
\( \lambda = 1 \)
\(\{0, 1, 2, \ldots\}\)
\(f(k) = e^{-\lambda} \frac{\lambda^k}{k!}, \; k \in \{0, 1, \ldots\} \)
\(\lfloor\lambda\rfloor\) if \( \lambda \) is not an integer; \( \lambda \) and \( \lambda - 1 \) if \( \lambda \) is a positive integer
\(F(x) = \frac{1}{x!} \gamma(x + 1, \lambda), \; x \in \{0, 1, 2, \ldots\}\) where \(\gamma\) is the lower incomplete gamma function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(G(t) = e^{\lambda (t - 1)}, \; t \in (-\infty, \infty)\)
\(M(t) = \exp\left(\lambda(e^t - 1)\right), \; t \in (-\infty, \infty)\)
\(\varphi(t) = \exp\left(\lambda(e^{i t} - 1)\right), \; t \in (-\infty, \infty)\)
\(m(k) = \lambda^k, \; k \in \{0, 1, 2, \ldots\}\)
\(\lambda\)
\(\lambda\)
\(\frac{1}{\sqrt{\lambda}}\)
\(\frac{1}{\lambda}\)
\(\lambda [1 - \log(\lambda)] + e^{-\lambda} \sum_{k=0}^\infty \frac{\lambda^k \log(k!)}{k!}\)
\(Q\left(\frac12\right) \approx \lfloor\lambda + 1/3 - 0.02/\lambda\rfloor\)
\(Q\left(\frac14)\right)\) where \(Q\) is the quantile function
\(Q\left(\frac34\right)\) where \(Q\) is the quantile function
exponential
power series
The Poisson distribution is named for Simeon Poisson who first used the distribution in 1838 in a study of judgements in court cases
consul1973generalization
Rademacher distribution
discrete
The Rademacher distribution arises in physics and in bootstrapping
\(k\in\{-1,1\}\,\)
N/A
\(Q(p) = -1, \; p \in [0, \frac{1}{2}]; \quad Q(p) = 1, \; p \in (\frac{1}{2}, 1]\)
\(M(t) = \cosh(t), \; t \in (-\infty, \infty)\)
\(M(t) = \cos(t), \; t \in (-\infty, \infty)\)
\(\mu(n) = 1, \; n \in \{0, 2, \ldots\}; \quad \mu(n) = 0, \; n \in \{1, 3, \ldots\}\)
\(0\,\)
\(1\,\)
\(0\,\)
\(-2\,\)
\(\ln(2)\)
\(0\,\)
\(-1\)
\(1\)
The Rademacher distribution is named for the German mathematician Hans Rademacher
montgomery1990distribution
Rayleigh distribution
continuous
The Rayleigh distribution governs the magnitude of a vector with independent, normal components that have zero mean and the same variance
\(\sigma \in (0, \infty)\), scale
\( \sigma = 1 \)
\([0, \infty)\)
\(f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2 \sigma^2}\right), \; x \in [0, \infty) \)
\(\sigma\)
\(F(x) = 1 - \exp\left(-\frac{x^2}{2 \sigma^2}\right), \; x \in [0, \infty)\)
\(Q(p) = \sigma \sqrt{-2 \ln(1 - p)}, \; p \in [0, 1)\)
\(M(t) = 1 + \sqrt{2 \pi} \sigma t \exp\left(\frac{\sigma^2 t^2}{2}\right) \Phi(t), \; t \in (-\infty, \infty)\) where \(\Phi\) is the standard normal distribution function
\(\mu(n) = \sigma^n 2^{n/2} \Gamma\left(1 + \frac{n}{2}\right), \; n \in [0, \infty)\), where \(\Gamma\) denotes the gamma function
\(\sigma \sqrt{\frac{\pi}{2}}\)
\(\sigma^2 (2 - \pi / 2)\)
\(\frac{2 \sqrt{\pi} (\pi - 3)}{(4 - \pi)^{3/2}}\)
\(-\frac{6 \pi^2 - 24 \pi + 16}{(4 - \pi)^2}\)
\(1 + \ln\left(\frac{\sigma}{\sqrt{2}}\right) + \frac{\gamma}{2}\) where \(\gamma\) is Euler's constant
\(\sigma \sqrt{\ln(4)}\,\)
\(\sigma \sqrt{\ln(16) - \ln(9)}\)
\(\sigma \sqrt{\ln(16)}\)
scale
The Rayleigh distribution is named for the English mathematician Lord Rayleigh (John William Strutt)
kuruoglu2004modeling
Rice distribution
Rician distribution
continuous
The Rice distribution governs the magnitude of a circular bivariate normal random vector
\(\nu \in [0, \infty)\), the distance parameter
\(\sigma \in (0, \infty)\), the scale parameter
\([0, \infty)\)
\(\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)\)
\(F(x) = 1 - Q\left(\frac{\nu}{\sigma}, \frac{x}{\sigma}\right), \; x \in [0, \infty)\) where \(Q\) is the Marcum \(Q\)-function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(\sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)\)
\(2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)\)
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
scale
The Rice distribution is named for Stephen O. Rice who used the distribution in 1945 in his study of random noise
rice1945mathematical
semicircle distribution
Wigner distribution
Stato-Tate distribution
continuous
The semicircle distribution arises as the limiting distribution of the eigenvalues of random symmetric matrices
\(r \in (0, \infty)\), scale (radius)
\( a \in (-\infty, \infty) \), location (center)
\( a = 0, \; r = 1 \)
\( [a - r, a + r] \)
\(f(x) = \frac{2}{\pi r^2} \sqrt{r^2 - (x - a)^2}, \; x \in [a - r, a + r]\)
\( a \)
\( F(x) = \frac{1}{2} + \frac{x - a}{\pi r^2} \sqrt{r^2 - (x - a)^2} + \frac{1}{\pi} \arcsin\left(\frac{x - a}{r}\right), \; x \in [a - r, a + r] \)
\(Q(p) = F^{-1}(p), \; p \in [0, 1]\) where \(F\) is the distribution function
\(M(t) = 2 e^{t a} \frac{I_1(r t)}{r t}, \; t \in (-\infty, \infty)\) where \(I_1\) is the modified Bessel function
\(\varphi(t) = 2 e^{i t a} \frac{J_1(r t)}{r t}, \; t \in (-\infty, \infty)\) where \(J_1\) is the Bessel function
For \( a = 0 \), \( \mu(2 n) = \left(\frac{r}{2}\right)^{2 n} \frac{1}{n + 1} \binom{2n}{n} \) and \( \mu(2 n + 1) = 0 \) for \( n \in \{0, 1, \ldots\} \)
\( a \)
\(\frac{r^2}{4}\)
\(0\,\)
\(-1\,\)
\(\ln(\pi r) - \frac{1}{2}\)
\( a\,\)
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
location
scale
The semicircle distribution was used by the physicist Eugene Wigner in the study of random matrices. The distribution was also used by Nikio Sato and John Tate in a conjecture in number theory
abramowitz1972handbook
triangular distribution
triangle distribution
continuous
The triangular distribution arises from various simple combinations of continuous uniform distributions
\(a \in (-\infty, \infty)\), location, the left endpoint
\( w \in (0, \infty) \), scale, the width of the interval
\( p \in [0, 1] \), shape
\(b = a + w\), the right endpoint
\(c = a + p w\), the location of the vertex
\( a = 0, \; w = 1, \; p = \frac{1}{2} \)
\([a, a + w]\)
\(f(x) = \begin{cases} \frac{2(x - a)}{(b - a)(c-a)} & a \le x \leq c, \\ \frac{2(b - x)}{(b - a)(b - c)} & c \lt x \le b \end{cases} \)
\( c \)
\(F(x) = \begin{cases} \frac{(x - a)^2}{(b - a)(c - a)} & a \le x \leq c, \\ 1 - \frac{(b - x)^2}{(b - a)(b - c)} & c \lt x \le b \end{cases}\)
\(Q(u) = \begin{cases} a + \sqrt{(b - a)(c - a) u}, & \; u \in [0, (c - a) / (b - a)] \\ b - \sqrt{(1 - u)(b - a)(b - c)}, & u \in [(c - a) / (b - a), 1] \end{cases}\)
\(M(t) = 2 \frac{(b - c) e^{a t} - (b - a) e^{c t} + (c - a) e^{b t}}{(b - a)(c - a)(b - c) t^2}, \; t \in (-\infty, \infty)\)
\(\frac{a + b + c}{3}\)
\(\frac{a^2 + b^2 + c^2 - a b - a c - b c}{18}\)
\(\frac{\sqrt2(a + b - 2 c)(2 a - b - c)(a - 2 b + c)}{5 (a^2 + b^2 + c^2 - a b - a c - b c)^{3/2}}\)
\(-\frac{3}{5}\)
\(\frac{1}{2} + \ln\left(\frac{b - a}{2}\right)\)
\(\begin{cases} a + \frac{\sqrt{(b - a)(c - a)}}{\sqrt{2}}, & c \ge \frac{a + b}{2}, \\ b - \frac{\sqrt{(b - a)(b - c)}}{\sqrt{2}}, & c \le \frac{a + b}{2} \end{cases}\)
\(\begin{cases} a + \sqrt{\frac{1}{4}(b - a)(c - a)}, & c \geq \frac{3}{4} a + \frac{1}{4} b \\ b - \sqrt{\frac{3}{4}(b - a)(b - c)} & c \leq \frac{3}{4} a + \frac{1}{4} b \end{cases}\)
\(\begin{cases} a + \sqrt{\frac{3}{4}(b - a)(c - a)}, & c \geq \frac{1}{4} a + \frac{3}{4} b \\ b - \sqrt{\frac{1}{4}(b - a)(b - c)}, & c \leq \frac{1}{4} a + \frac{3}{4} b \end{cases}\)
location
scale
ren2002novel
U-quadratic distribution
continuous
the U-quadratic distribution models certain symmetric, bimodal variables
\(c \in (-\infty, \infty)\), location, center
\( w \in (0, \infty) \), scale, radius
\(a = c - w\), the left endpoint
\(b = c + w \), the right endpoint
\( c = 0, \; w = 1 \)
\([a, b]\)
\(f(x) = \frac{3}{2 w} \left(\frac{x - c}{w}\right)^2, \; x \in [a, b]\)
\(\{a, b\}\)
\( F(x) = \frac{1}{2}\left[1 + \left(\frac{x - c}{w}\right)^3\right], \; x \in [a, b] \)
\(Q(p) = c + w(2 p - 1)^{1/3}, \; p \in [0, 1]\)
\(c = \frac{a + b}{2}\)
\(\frac{3}{5} w^2 = \frac{3}{5}(b - a)^2\)
\(0\)
\(-\frac{38}{21}\)
\(c = \frac{a + b}{2}\)
\( c - \frac{1}{2^{1/3}} w \)
\( c + \frac{1}{2^{1/3}} w \)
location
scale
rubinstein1973comparative
von Mises distribution
circular normal distribution
Tikhanov distribution
continuous
The von Mises distribution is used as an approximation to the wrapped normal distribution
\(\mu \in (-\infty, \infty)\), the location parameter
\(\beta \in (0, \infty)\), the concentration parameter
any interval of length \( 2 \pi \)
\(\frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}\)
\(\mu\)
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(\mu\)
\(\textrm{var}(x)=1-I_1(\kappa)/I_0(\kappa)\)(circular)
\(-\beta \frac{I_1(\beta)}{I_0(\beta)} + \ln[2 \pi I_0(\beta)]\) where \(I_n\) is the modfied Bessel function of order \(n\)
\(\mu\)
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
location
The von Mises distribution is named for Richard von Mises based on his work in diffusion processes
abramowitz1972handbook
Wald distribution
inverse Gaussian distribution
continuous
The Wald distribution governs the time that Brownian Motion with positive drift reaches a fixed positive value
\(\mu \in (0, \infty)\), the mean
\(\lambda \in (0, \infty)\), the shape parameter
\( \mu = 0, \; \lambda = 1 \)
\(x\in(0,\infty)\)
\(\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp{\frac{-\lambda(x-\mu)^2}{2\mu^2x}}\)
\(\mu\left[\left(1+\frac{9\mu^2}{4\lambda^2}\right)^\frac{1}{2}-\frac{3\mu}{2\lambda}\right]\)
\(\Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}-1\right)\right)\)\(+\exp\left(\frac{2\lambda}{\mu}\right)\Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1\right)\right)\) where \(\Phi\left(\right)\) is the standard normal distribution function.
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(M(t) = \exp \left[ \frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2 \mu^2}{\lambda} t} \right)\right], \; t \in (-\infty, \frac{\lambda}{2 \mu^2})\)
\(\varphi(t) = \exp \left[ \frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2 \mu^2}{\lambda} i t} \right)\right], \; t \in (-\infty, \infty)\)
\(\mu\)
\(\frac{\mu^3}{\lambda}\)
\(3\left(\frac{\mu}{\lambda}\right)^{1/2}\)
\(\frac{15\mu}{\lambda}\)
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
The Wald distribution is named for Abraham Wald
chhikara1988inverse
Weibull distribution
continuous
The Weibull distribution is used to model the failure times
\(k \in (0, \infty)\), the shape parameter
\(\lambda \in (0, \infty)\), the scale parameter
\( k = 1, \; \lambda = 1 \)
\( (0, \infty) \)
\(f(x) =\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} \exp\left[-\left(\frac{x}{\lambda}\right)^{k}\right], \; x \in (0, \infty)\)
0 if \( k = 1 \), \(\lambda\left(\frac{k-1}{k}\right)^{\frac{1}{k}}\) if \( k \gt 1 \)
\(F(x) = 1 - \exp\left[-\left(\frac{x}{\lambda}\right)^k\right], \; x \in (0, \infty)\)
\(Q(p) = \lambda \left[- \ln(1 - p)\right]^{1/k}, \; p \in (0, 1)\)
\(M(t) = \sum_{n=0}^\infty \frac{t^n \lambda^n}{n!} \Gamma\left(1 + \frac{n}{k}\right), \; t \in (-\infty, \infty), \; k \in (1, \infty)\) where \(\Gamma\) is the gamma function
\(\varphi(t) = \sum_{n=0}^\infty \frac{(i t)^n \lambda^n}{n!} \Gamma\left(1 + \frac{n}{k}\right), \; t \in (-\infty, \infty)\)
\(\lambda \Gamma(1 + 1/k)\)
\(\lambda^2 \Gamma(1 + 2/k) - \mu^2\) where \( \mu \) is the mean
\(\frac{\Gamma(1 + 3/k)\lambda^3 - 3\mu \sigma^2 - \mu^3}{\sigma^3}\) where \( \mu \) is the mean and \( \sigma \) is the standard deviation
\( \gamma_{2}=\frac{\lambda^4\Gamma(1+\frac{4}{k})-4\gamma_{1}\sigma^3\mu-6\mu^2\sigma^2-\mu^4}{\sigma^4}-3 \)
\(\lambda[\ln(2)]^{1/k}\)
\(\lambda [\ln(4) - \ln(3)]^{1/k}\)
\(\lambda [\ln(4)]^{1/k}\)
exponential
scale
The Weibull distribution is named for Waloddi Weibull who published a paper on the distribution in 1951. The distribution was used earlier by Maurice Frechet. The term Weibull distribution was first used in 1955 in a paper by Julius Lieblein
weibull1951statistical
zeta distribution
Zipf distribution
discrete
The zeta distribution models ranks and sizes of certain randomly chosen items
\(s \in [1, \infty)\), shape
\( s = 1 \)
\(\{1, 2, \ldots\}\)
\(f(k) = \frac{1}{\zeta(s) k^s}, \; k \in \{1, 2, \ldots\}\) where \( \zeta \) is the zeta function
\(1\)
\(F(k) = \frac{H_{k,s}}{\zeta(s)}, \; k \in \{1, 2, \ldots\}\), where \(H_{k,s}\) is a generalized harmonic number.
\(Q(p) = F^{-1}(p), \; p \in [0, 1)\) where \(F\) is the distribution function
\(\frac{\zeta(s - 1)}{\zeta(s)}\) for \(s \gt 2\)
\(\frac{\zeta(s)\zeta(s - 2) - \zeta(s - 1)^2}{\zeta(s)^2}\) for \(s \gt 3\)
\( Q\left(\frac{1}{2}\right) \) where \( Q \) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function
The Zipf distribution is named for the American linguist George Kingsley Zipf, who studied the distribution in the context of the frequency of words
johnson2005univariate
location-scale distribution
continuous
Location scale distributions correspond to linear transformations (with positive slope) of a basic random variable, and often correspond to a change of units in a physical problem
the standard distribution, a continuous distribution with support on an interval \(S_0\)
\(\mu \in (-\infty, \infty)\), the location parameter
\(\sigma \in (0, \infty)\), the scale parameter
\(S = \{\mu + \sigma x: x \in S_0\}\)
\(f(x) = \frac{1}{\sigma} f_0\left(\frac{x - \mu}{\sigma}\right), \; x \in S\) where \(f_0\) is the probability density function of the standard distribution
\(\mu + \sigma x_0\) where \(x_0\) is a mode of the standard distribution
\(F(x) = F_0\left(\frac{x - \mu}{\sigma}\right), \; x \in S\) where \(F_0\) is the distribution function of the standard distribution
\(Q(p) = \mu + \sigma Q_0(p), \; p \in (0, 1)\) where \(Q_0\) is the quantile function of the standard distribution
\(M(t) = e^{\mu t} M_0(\sigma t)\) where \(M_0\) is the moment generating function of the standard distribution
\(\varphi(t) = e^{i \mu t} \varphi_0(\sigma t)\) where \(\phi\) is the characteristic function of the standard distribution
\(m(n) = \sum_{i=0}^n {n \choose i} \sigma^i \mu^{n-i} m_0(i), \; n \in \{1, 2, \ldots\}\) where \(m_0(i)\) is the \(i\)th raw moment of the standard distribution
\(\mu + \sigma \mu_0\) where \(\mu_0\) is the mean of the standard distribution
\(\sigma^2 \sigma_0^2\) where \(\sigma_0^2\) is the variance of the standard distribution
\(\gamma_{0,1}\) where \(\gamma_{0,1}\) is the skewness of the standard distribution
\(\gamma_{0,2}\) where \(\gamma_{0,2}\) is the kurtosis of the standard distribution
\(\ln(\sigma) + I_0\) where \(I_0\) is the entropy of the standard distribution
\(\mu + \sigma q_{0,2}\) where \(q_{0,2}\) is the median of the standard distribution
\(\mu + \sigma q_{0,1}\) where \(q_{0,1}\) is the first quartile of the standard distribution
\(\mu + \sigma q_{0,3}\) where \(q_{0,3}\) is the third quartile of the standard distribution
meyer1987two
folded normal distribution
continuous
The folded normal distribution governs \(\left|X\right|\) when \(X\) has a normal distribution
\(\mu \in (-\infty, \infty)\), the normal mean
\(\sigma \in (0, \infty)\), the normal standard deviation
\( \mu = 0, \; \sigma = 1 \)
\([0, \infty)\)
\(f(x) = \frac{1}{\sigma \sqrt{2\pi}} \, \exp \left( -\frac{(-x - \mu)^2}{2\sigma^2} \right) + \frac{1}{\sigma \sqrt{2 \pi}} \, \exp \left(-\frac{(x - \mu)^2}{2 \sigma^2} \right), \; x \in [0, \infty) \)
\( F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right) + \Phi\left(\frac{x + \mu}{\sigma}\right) - 1, \; x \in [0, \infty) \) where \( \Phi \) is the standard normal distribution function
\(F^{-1}(p), p \in (0, 1)\) where \(F\) is the distribution funciton
\(\sigma \sqrt{\frac{2}{\pi}} \exp\left(-\frac{\mu^2}{2 \sigma^2} \right) + \mu \left[1 - 2 \Phi\left(-\frac{\mu}{\sigma}\right) \right]\) where \(\Phi\) is the standard normal distribution function
\( \mu^2 + \sigma^2 - \left\{ \sigma \sqrt{2/\pi} \exp(-\mu^2 / 2 \sigma^2) + \mu\left[1 - 2 \Phi(-\mu / \sigma)\right] \right\}^2 \) where \( \Phi \) is the standard normal distribution function
\(Q(\frac{1}{2})\) where \(Q\) is the quantile function
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the qunatile function
leone1961folded
half normal distribution
continuous
The half normal distribution governs \(|X|\) when \(X\) has a normal disstribution with mean 0
\(\sigma \in (0, \infty) \), the scale parameter
\( \sigma = 1 \)
\([0, \infty)\)
\(f(x) = \frac{\sqrt{2}}{\sigma \sqrt{\pi}} \exp \left(-\frac{x^2}{2 \sigma^2} \right), \; x \in (0, \infty)\)
0
\(\sigma \sqrt{\frac{2}{\pi}}\)
\( \sigma^2 \left(1 - \frac{2}{\pi}\right) \)
\( \frac{1}{2} \ln \left( \frac{ \pi \sigma^2 }{2} \right) + \frac{1}{2}\)
\(\mu(n) = \frac{\pi^{(n - 1) / 2}}{\sigma^n} \Gamma\left(\frac{1}{2}(n + 1) \right)\) where \(\Gamma\) is the gamma function
\(\frac{\sqrt{2}(4 - \pi)}{(\pi - 2)^{3/2}}\)
\(\frac{8(\pi - 3)}{(\pi - 2)^2}\)
scale
pescim2010beta
birthday distribution
occupancy distribution
discrete
This distribution models the number of empty cells when \(n\) balls are distributed at random into \(m\) cells
\(m \in \{1, 2, \ldots\}\), the number of cells
\(n \in \{1, 2, \ldots\}\), the number of balls
\(\{\max\{m-n, 0\}, \ldots, m - 1\}\)
\(f(x) = \binom{m}{x} \sum_{j=0}^{m-x} (-1)^j \binom{m - x}{j} \left(1 - \frac{x + j}{m}\right)^n, \quad x \in \{\max\{m-n,0\}, \ldots, m-1\}\)
\(F(x) = \sum_{j = 0}^x f(j), \quad x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function
\(\mu_{(k)} = \frac{m!}{(m - k)!} \left(\frac{m - k}{m} \right)^n, \quad k \in \{1, 2, \ldots\}\)
\(G(t) = \sum_{k=0}^m \binom{m}{k} \left(\frac{m - k}{m}\right)^n (t - 1)^k, \quad t \in R\)
\(m \left(1 - \frac{1}{m}\right)^n\)
\(m (m - 1) \left(1 - \frac{2}{m}\right)^n + m \left(1 - \frac{1}{m}\right)^n - m^2 \left(1 - \frac{1}{m}\right)^{2n}\)
\(\frac{\mu_3 - 3 \mu_1 \mu_2 + 2 \mu_1^2}{\sigma^3}\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation
\(\frac{\mu_4 - 4 \mu_1 \mu_3 + 6 \mu_1^2 -3 \mu_1^4}{\sigma^4} - 3\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation
\(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function
\(Q(\frac{1}{2})\) where \(Q\) is the quantile function
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the qunatile function
\(H = -\sum_{x=0}^n \log[f(x)] f(x)\) where \(f\) is the probability density function
munford1977note
matching distribution
discrete
The matching distribution governs the number of matches in a random permutation of \(\{1, 2, \ldots, n\}\)
\(n \in \{2, 3, \ldots\}\), the number of objects permuted
\(\{0, 1, \ldots, n\}\)
\(f(x) = \frac{1}{x!} \sum_{j=0}^{n - x} \frac{(-1)^j}{j!}, \; x \in \{0, 1, \ldots, n\}\)
\(0\) if \(n\) is even; \(1\) if \(n\) is odd
\(F(x) = \sum_{j = 0}^x f(j), \; x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function
\(\mu_{(k)} = 1, \; k \in \{1, 2, \ldots, n\}; \quad \mu_{(k)} = 0, \; k \in \{n + 1, n + 2, \ldots\}\)
\(G(t) = \sum_{k=1}^n \frac{(t-1)^k}{k!}, \; t \in R\)
\(1\)
\(1\)
\(\frac{\mu_3 - 3 \mu_1 \mu_2 + 2 \mu_1^2}{\sigma^3}\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation
\(\frac{\mu_4 - 4 \mu_1 \mu_3 + 6 \mu_1^2 -3 \mu_1^4}{\sigma^4} - 3\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation
\(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function
\(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function
\(Q\left(\frac{3}{4}\right)\) where \(Q\) is the qunatile function
\(H = -\sum_{x=0}^n \log[f(x)] f(x)\) where \(f\) is the probability density function
The matching problem was first formulated by Pierre-Redmond Montmort
coupon-collector distribution
discrete
This distribution models the number number of samples needed to obtain \(k\) distinct values when sampling at random, with replacement from a population of \(m\) objects
\(m \in \{1, 2, \ldots\}\), the population size
\(k \in \{1, 2, \ldots\}\), the number of distinct values to be obtained
\(\{k, k + 1, \ldots\}\)
\(f(x) = \binom{m - 1}{k - 1} \sum_{j=0}^{k-1} \binom{k-1}{j} \left(\frac{k - j - 1}{m}\right)^{x-1}, \quad x \in \{k, k + 1, \ldots\}\)
\(F(x) = \sum_{j = 0}^x f(j), \quad x \in \{k, k + 1, \ldots\}\) where \(f\) is the probability density function
\(G(t) = \prod_{i=1}^k \frac{m - (i - 1)}{m - (i - 1)t}, \quad |t| \lt \frac{m}{k - 1} \)
\(\sum_{i=1}^k \frac{m}{m - i + 1}\)
\(\sum_{i=1}^k \frac{(i-1)m}{(m - i + 1)^2}\)
motwani1995randomized
finite order statistic distribution
discrete
This distribution models an order statistic when a sample is chosen at random, without replacement, from a finite, ordered population
\(m \in \{1, 2, \ldots\}\), the population size
\(n \in \{1, 2, \ldots, m\}\), the sample size
\(k \in \{1, 2, \ldots, n\}\), the the order
\(\{k, k + 1, \ldots, m - n + 1\}\)
\(f(x) = \frac{\binom{x-1}{k-1} \binom{m-x}{n-k}}{\binom{m}{n}}, \quad x \in \{k, k + 1, \quad m - n + 1\}\)
\(k \frac{m+1}{n+1}\)
\(k(n - k + 1) \frac{(m + 1)(m - n)}{(m + 1)^2 (n + 2)}\)
Erlang distribution
continuous
The Erlang probability distribution is related exponential and Gamma distributions
and is used to examine the number of event arrivals. For instance telephone calls which
might be made at the same time to the operators of the switching stations. This work
on telephone traffic engineering has been expanded to consider waiting times in queueing
systems in general
\(k \in \mathbb{N}\), shape parameter
\(\lambda > 0\), rate parameter
\(\theta = 1/\lambda > 0\), scale parameter
\(\scriptstyle x\;\in\;[0,\,\infty)\!\)
\(\scriptstyle\frac{\lambda^k x^{k-1}e^{-\lambda x}}{(k-1)!\,}\)
\(\scriptstyle\frac{\gamma(k,\,\lambda x)}{(k\,-\,1)!}\;=\;1\,-\,\sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}\)
\(\scriptstyle\frac{k}{\lambda}\,\)
No simple closed form
\(\scriptstyle\frac{k}{\lambda^2}\,\)
angusintroduction
Generalized Gamma distribution
continuous
The generalized gamma distribution is not often used to model life data by itself, but it is sometimes used to determine which of those life distributions should be used to model a particular set of data
\(\alpha \in (0, \infty)\), the scale parameter
\(\beta \in (0, \infty)\), the shape parameter
\(\delta \in (0, \infty)\), the shape parameter
\((0, \infty)\)
\(f(x; \alpha, \beta, \delta) = \frac{(\delta/\alpha^\beta) x^{\beta-1} e^{-(x/\alpha)^\delta}}{\Gamma(\beta/\delta)}\), where \(\Gamma\) is the gamma function.
\(F(x; \alpha, \beta, \delta) = \frac{\gamma(\beta/\delta, (x/\alpha)^\delta)}{\Gamma(\beta/\delta)}\), where \(\gamma\) denotes lower incomplete gamma function
stacy1962generalization
Makeham distribution
Gompertz-Makeham law of mortality
continuous
The Gompertz-Makeham law states that the death rate is the sum of an age-independent component and an age-dependent component, which increases exponentially with age
\(\gamma \in (0, \infty)\)
\(\delta \in (0, \infty)\)
\(\kappa \in (0, \infty)\), all the parameters are in units reciprocal of the units of the variable, \(x\)
\((0, \infty)\)
\(f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta(\kappa^x-1)}{log(\kappa)})\)
\(1-\exp(-\frac{\gamma x \log(\kappa) + \delta \kappa^x -\delta}{\log(\kappa)})\)
gavrilov1983human
hypoexponential distribution
generalized Erlang distribution
continuous
The hypoexponential distribution has a coefficient of variation less than one, compared to the hyperexponential distribution,
which has a coefficient of variation greater than one and the exponential distribution which has a coefficient of variation of one
\(\{\lambda_1,...,\lambda_n\}\),\(\lambda_i \in (0, \infty), \lambda_i \neq \lambda_j\) for \(i \neq j\), rate parameters
\([0, \infty)\)
\(f(x) = \sum_{i=1}^{n}(\lambda_i)exp(-x/\lambda_i)(\prod_{j=1,j\neq i}^{n}\frac{\lambda_i}{\lambda_i-\lambda_j})\)
Expressed as a phase-type distribution: \(1-\boldsymbol{\Lambda}e^{x\Theta}\boldsymbol{1}\), where
\(\Theta=\Theta(\lambda_1,...,\lambda_n) \) is a phase-type matrix, and \( \alpha = (1,0,0,...,0)\)
\(\boldsymbol{\alpha}(tI-\Theta)^{-1}\Theta\mathbf{1}\)
\(\sum^{k}_{i=1}1/\lambda_{i}\,\)
\((k-1)/\lambda\) if \(\lambda_{k} = \lambda\)
\(\sum^{k}_{i=1}1/\lambda^2_{i}\)
\(\ln(2)\sum^{k}_{i=1}1/\lambda_{i}\)
\(2(\sum^{k}_{i=1}1/\lambda_{i}^3)/(\sum^{k}_{i=1}1/\lambda_{i}^2)^{3/2}\)
bolch2006queueing
doubly noncentral t distribution
continuous
The doubly noncentral t distribution is an extended version of the singly noncentral t distribution in that it has two noncentrality parameters instead of just one
See http://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.1969.tb00102.x/pdf
krishnan1968series
hyperexponential distribution
continuous
The hyperexponential distribution has a coefficient variation greater than one, compared to the hypoexponential distribution which has a coefficient variation less than one and the exponential distribution which has a coefficient variation of one
\(\alpha_1,...,\alpha_n \in (0, \infty), \alpha_i \neq \alpha_j for i \neq j\)
\(p_i > 0, \sum_{i=1}^{n} p_i = 1\)
\((0, \infty)\)
\(f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}\)
singh2007estimation
Muth distribution
continuous
The Muth distribution is related to reliability models. It has mostly
a theoretical interest. Muth distribution has two basic properties: (i) the
mode of this random model is a function involving the golden ratio and (ii)
the second non-central moment can be expressed in terms of the exponential
integral function. The moments of higher order cannot be expressed in a simple way.
The Muth distribution does not have the variate generation property for
simulation purposes. Its quantile function can be expressed in closed form in
terms of the negative branch of the Lambert W function. The limit distributions
of the maxima and minima of the Muth distribution are the Gumbel and Weibull
distributions, respectively
\(\kappa \in [0, 1]\), the shape parameter
\((0, \infty)\)
\(f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}\)
muth1960optimal
generalized error distribution
generalized normal distribution
generalized Gaussian distribution
exponential power distribution
continuous
The error distribution is a parametric family of symmetric distributions. It adds a shape parameter to the normal distribution
\(a \in (-\infty, \infty)\), the mean
\(b \in (0, \infty)\), the scale parameter
\(c \in (0, \infty)\), the shape parameter
\((-\infty, \infty)\)
\(f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}\), where \(\Gamma\) denotes the gamma function.
hosking2005regional
minimax distribution
continuous
The Minimax distribution is an alternative two-parameter distribution of the Beta distribution
\(\beta \in (0, \infty)\)
\(\gamma \in (0, \infty)\)
\((0, 1)\)
\(f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}\)
marchand2002minimax
noncentral F distribution
continuous
The noncentral F distribution is a generalization of the ordinary F distribution
\(\delta \in (0, \infty)\), noncentrality parameter
\(n_1 \in (1,2, 3, ...)\), degrees of freedom of numerator
\(n_2 \in (1,2, 3, ...)\), degrees of freedom of denominator
\((0, \infty)\)
\(f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}
e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}\)
where \(\Gamma\) denotes the gamma function
increasing-decreasing-bathtub distribution
continuous
The IDB distribution can be used to model either an increasing, decreasing or bathtub shaped failure rate function,
which is a combination of a linearly increasing failure rate and a decreasing failure rate function
\(\delta \in (0, \infty)\)
\(\kappa \in (0, \infty)\)
\(\gamma \in [0, \infty)\)
\((0, \infty)\)
\(f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}\)
2\\
\mbox{Does not exist}
&\nu_2\le2\\
\end{cases}\)
]]>
4\\
\mbox{Does not exist}
&\nu_2\le4.\\
\end{cases}\)
]]>
kay1993fundamentals
Benford's law
first digit law
significant digit law
discrete
Benford's law states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way
\(b \in \{2, 3, \ldots\}\), the base
\(b = 10\)
\(\{1, 2, \ldots, b - 1\}\)
\(f(x) = \log_b(x + 1)- \log_b(x) = \log_b\left(\frac{x + 1}{x}\right), \; x \in \{1, 2, \ldots, b - 1\}\)
hill1995statistical
doubly noncentral F distribution
continuous
The doubly noncentral F distribution is an extended version of the singly noncentral F distribution in that it has two noncentrality parameters instead of just one.
In other words, the doubly noncentral F-distribution describes the distribution \(\frac{X/n_1}{Y/n_2}\) for
two independently distributed noncentral chi-squared variables \(X \sim chi_{n_1}^2(\lambda_1)\) and
\(Y \sim chi_{n_2}^2(\lambda_2)\)
\(\delta \in (0, \infty)\)
\(\gamma \in (0, \infty)\)
\((0, \infty)\)
\(f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}\), w
where \(n_1\) and \(n_2\) are the degrees of freedom of the numerator and denominator, and \(B\) is the beta function
bulgren1971representations
two-sided power distribution
continuous
The two-sided power distribution is an alternative to the triangular distribution, allowing for a nonlinear distribution. Triangular and uniform distributions are special cases of the two-sided power distribution
\(n \in (0, \infty)\)
\(a \in (-\infty, \infty)\)
\(b \in (a, \infty)\)
\(m=(b-a)\theta+a\)
\((a, b)\)
\(f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a\lt x\le m \\ \frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x\lt b \end{cases}\)
van2002standard
extreme value distribution
Gumbel distribution
continuous
The Extreme Value distribution is the limiting distribution of the minimum of a large number of unbounded identically distributed random variables
\(a \in (-\infty, \infty)\), location
\(b \in (0, \infty)\), scale
\( a = 0, \; b = 1 \)
\((-\infty, \infty)\)
\( f(x) = \frac{1}{b} \exp\left(-\frac{x - a}{b}\right) \exp\left[-\exp\left(-\frac{x - a}{b}\right)\right], \; x \in (-\infty, \infty) \)
\( a \)
\( F(x) = \exp\left[-\exp\left(-\frac{x - a}{b}\right)\right], \; x \in (-\infty, \infty) \)
\( F^{-1}(p) = a - b \ln[-\ln(p)], \; p \in (0, 1) \)
\( M(t) = e^{a t} \Gamma(1 - b t), \; t \in (-\infty, 1/b) \) where \( \Gamma \) is the gamma function
\( a + b \gamma \) where \( \gamma \) is Euler's constant
\( b^2 \pi / 6 \)
\( 12 \sqrt{6} \zeta(3) / \pi^3 \), where \(\zeta\) is the Riemann zeta function
\( 12/5 \)
\( a - b \ln[\ln(2)] \)
\(a - b \ln[\ln(4) - \ln(3)]\)
\(a - b \ln[\ln(4)]\)
\( \ln(b) + \gamma + 1 \)
location
scale
The Gumbel distribution is named for Emil Gumbel, who derived it in his study of extreme values in 1954
embrechts2011modelling
Lomax distribution
Pareto Type II distribution
continuous
The Lomax distribution is essentially a Pareto distribution that has been shifted so that its support begins at zero
\(\alpha \in (0, \infty)\), the shape parameter
\(\lambda \in (0, \infty)\), the scale parameter
\(x\ge0\)
\({\alpha\over\lambda}\left[{1+{x\over\lambda}}\right]^{-(\alpha+1)}\)
\(1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}\)
\(\log(\sigma)\,+\,\gamma\xi\,+\,(\gamma+1)\)
coles2001introduction
generalized Pareto distribution
continuous
The generalized Pareto distribution allows a continuous range of possible shapes that includes both the exponential and Pareto distributions as special cases
\(\kappa \in (-\infty, \infty)\), the shape parameter
\(\sigma \in (0, \infty)\), the scale parameter
\(\mu \in (0, \infty)\), the location parameter
\(\mu\leqslant x\leqslant\mu-\sigma/\kappa\,\;(\kappa<0)\)
]]>
\(\frac{1}{\sigma}(1+\kappa z)^{-(1/\kappa+1)}\), where \(z=\frac{x-\mu}{\sigma}\)
\(1-(1+\kappa z)^{-1/\kappa}\,\)
\(\mu+\frac{\sigma(2^{\kappa}-1)}{\kappa}\)
chotikapanich2008modeling
Kolmogorov-Smirnov test
Kolmogorov distribution
continuous
The Kolmogorov-Smirnov test can be modified to serve as a goodness of fit test
none
\((0, \infty)\)
\(f(x)=1-2[\exp{-x^2}-\exp{-4 x^2}+\exp{-9 x^2}-\exp{-16 x^2}+...]\)
kolmogorov1933sulla
logistic-exponential distribution
continuous
infinte
\( \alpha = \beta = \frac{1}{2} \)
lan2008logistic
power-function distribution
\(\alpha\), shape parameter \((>0)\)
\(\beta\), boundary parameters \( (\alpha \lt \beta) \)
continuous
nonsymmetric
finite
\( f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} \)
moothathu1986characterization
student t non-central distribution
continuous
nonsymmetric
infinite
\( f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx \)
2 ;\\
\mbox{Does not exist},
&\mbox{if }\nu\le2 .\\
\end{cases}\)
]]>
\(\sqrt{\frac{\nu}{2}}\frac{\Gamma\left(\frac{\nu+2}{2}\right)}{\Gamma\left(\frac{\nu+3}{2}\right)}\mu;\,\)
lenth1989algorithm
inverted gamma distribution
inverse gamma distribution
continuous
nonsymmetric
finite
positive
\(\alpha \gt 0 \), shape
\(\beta \gt 0 \), scale
\((0, \infty)\)
\( \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right) \)
\(\frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!\)
\(\frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)\)
\(\frac{\beta}{\alpha-1}\!\) for \(\alpha > 1\)
\(\frac{\beta}{\alpha+1}\!\)
\(\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!\) for \(\alpha > 2\)
\(\frac{4\sqrt{\alpha-2}}{\alpha-3}\!\) for \(\alpha > 3\)
\(\frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!\) for \(\alpha > 4\)
\(\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)\)
witkovsky2001computing
Fisher-Tippett distribution
continuous
nonsymmetric
finite
positive
\( \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right) \)
embrechts2011modelling
Gibrat's distribution
continuous
nonsymmetric
finite
positive
\( \frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right] \)
eeckhout2004gibrat
Gompertz distribution
continuous
nonsymmetric
finite
positive
\(\eta \in (0,\infty) \), shape paraemter
\(b \in (0,\infty) \), scale parameter
\(x \in [0,\infty)\)
\( b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right] \)
\(1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)\)
\(\text{E}\left(e^{-t x}\right)=\eta e^{\eta}\text{E}_{t/b}\left(\eta\right)\), \(\text{with E}_{t/b}\left(\eta\right)=\int_1^\infty e^{-\eta v} v^{-t/b}dv,\ t>0\)
\((-1/b)e^{\eta}\text{Ei}\left(-\eta\right)\), \(\text {where Ei}\left(z\right)=\int\limits_{-z}^{\infty}\left(e^{-v}/v\right)dv\)
\(\left(1/b\right)\ln\left[\left(-1/\eta\right)\ln\left(1/2\right)+1\right]\)
bemmaor2012modeling
multinomial distribution
discrete
nonsymmetric
finite
positive
\(X_i \in \{0,\dots,n\}\), where \(\Sigma X_i = n\!\)
\( f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k} \)
\(\biggl( \sum_{i=1}^k p_i e^{t_i} \biggr)^n\)
\(E\{X_i\} = np_i\)
\(\textstyle{\mathrm{Var}}(X_i) = n p_i (1-p_i)\), where \(\textstyle {\mathrm{Cov}}(X_i,X_j) = - n p_i p_j~~(i\neq j)\)
evans2000statistical
negative multinomial distribution
discrete
nonsymmetric
finite
positive
\( f(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} \)
\(\tfrac{k_0}{p_0}\,p\)
\(\tfrac{k_0}{p_0^2}\,pp' + \tfrac{k_0}{p_0}\,\operatorname{diag}(p)\)
gall2006modes
negative hypergeometric distribution
discrete
finite
positive
If \({x \choose 2} \lt\lt W\) and \({b \choose 2} \lt\lt B\) then \(X\) can be
approximated as a negative binomial random variable with parameters \(r = b\) and
\(p = \frac{W}{W+B}\). This approximation simplifies the distribution by looking as
a system with replacement for large values of \(W\) and \(B\)
\(W \in \{1,2,...\}\)
\(B \in \{1,2,...\}\)
\(b \in \{1,2,...,B\}\)
\(x=\{0,1,...,W\}\)
\( f(x) \frac{ { x+b-1 \choose x} {W+B-b-x \choose W-x} }{ {W+B \choose W} } \)
askey2010generalized
power series distribution
discrete
infinite
positive
\( f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \! \)
yanushauskas1980double
beta-Pascal distribution
discrete
infinite
positive
\( f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \! \)
johnson2005univariate
gamma-Poisson distribution
discrete
infinite
positive
\(\{k, k+1, \ldots\}\)
\(f(x) = {x-1 \choose k-1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}\)
\(\lfloor 1 + \frac{k-1}{p}\rfloor\)
\(F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}\) where \(f\) is the probability density function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(G(t) = \left[\frac{p t}{1 - (1-p) t}\right]^k, \; t \in (-\frac{1}{1-p}, \frac{1}{1-p})\)
\(M(t) = \left[\frac{p e^t}{1 - (1-p) e^t}\right]^k, \; t \in (-\infty, -\ln(1 - p))\)
\(\varphi(t) = \left[\frac{p e^{i t}}{1 - (1-p) e^{i t}}\right]^k, \; t\in (-\infty, \infty)\)
\(k \frac{1}{p}\)
\(k \frac{1-p}{p^2}\)
\(\frac{2-p}{\sqrt{k (1-p)}}\)
\(\frac{1}{k} \left[6 + \frac{p^2}{1 - p}\right]\)
\(Q(\frac{1}{2})\) where \(Q\) is the quantile function
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
The alternative name Pascal distribution is in honor of Blaise Pascal who used the distribution in his solution to the Problem of Points
el2006negative
Polya distribution
discrete
finite
positive
\(\{k, k+1, \ldots\}\)
\(f(x) = {x-1 \choose k-1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}\)
\(\lfloor 1 + \frac{k-1}{p}\rfloor\)
\(F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}\) where \(f\) is the probability density function
\(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function
\(G(t) = \left[\frac{p t}{1 - (1-p) t}\right]^k, \; t \in (-\frac{1}{1-p}, \frac{1}{1-p})\)
\(M(t) = \left[\frac{p e^t}{1 - (1-p) e^t}\right]^k, \; t \in (-\infty, -\ln(1 - p))\)
\(\varphi(t) = \left[\frac{p e^{i t}}{1 - (1-p) e^{i t}}\right]^k, \; t\in (-\infty, \infty)\)
\(k \frac{1}{p}\)
\(k \frac{1-p}{p^2}\)
\(\frac{2-p}{\sqrt{k (1-p)}}\)
\(\frac{1}{k} \left[6 + \frac{p^2}{1 - p}\right]\)
\(Q(\frac{1}{2})\) where \(Q\) is the quantile function
\(Q(\frac{1}{4})\) where \(Q\) is the quantile function
\(Q(\frac{3}{4})\) where \(Q\) is the quantile function
The alternative name Pascal distribution is in honor of Blaise Pascal who used the distribution in his solution to the Problem of Points
el2006negative
gamma-normal distribution
bivariate
continuous
infinite
\( f(x,\tau|\mu,\lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt{\lambda}}{\Gamma(\alpha)\sqrt{2\pi}} \, \tau^{\alpha-\frac{1}{2}}\,e^{-\beta\tau}\,e^{ -\frac{ \lambda \tau (x- \mu)^2}{2}} \)
\(\operatorname{E}(X)=\mu\,\! ,\quad \operatorname{E}(\tau)= \alpha \beta^{-1}\)
\(\operatorname{var}(X)= \frac{\beta}{\lambda (\alpha-1)} ,\quad
\operatorname{var}(\tau)=\alpha \beta^{-2}\)
bernardo2001bayesian
discrete Weibull distribution
discrete
infinite
positive
\( f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \! \)
englehardt2012methods
noncentral beta distribution
continuous
positive
\( f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1) \! \)
\(F(x) = \sum_{j=0}^\infty \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}B(x; a+j,b)\) where \(B\) is regularized incomplete beta function
abramowitz1972handbook
arctangent distribution
continuous
infinite
positive
\( f(x; \lambda, \phi)= \frac{\lambda}{[\arctan(\lambda \phi)+\pi/2]
[1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty
\lt \lambda \lt \infty) \! \)
pollastri2004some
log-gamma distribution
continuous
infinite
\( f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}\),
where \((-\infty \lt x \lt \infty) \! \)
demirhan2011multivariate
Bernoulli distribution
binomial distribution
If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in [0, 1]\) then \(Y = \sum_{i = 1}^n X_i\) has the binomial distribution with parameters \(n\) and \(p\).
convolution
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Bernoulli distribution
geometric distribution
If \((X_1, X_2, \ldots)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in (0, 1)\), then \(Y = \min\{n \in \{1, 2, \ldots\}: X_n = 1\}\) has the geometric distribution with parameter \(p\).
transformation
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Bernoulli distribution
negative binomial distribution
If \((X_1, X_2, \ldots)\) is a sequence of independent Bernoulli variables,
each with parameter \(p \in (0, 1)\),
then for \(ki \in \{1, 2, \ldots\}\), \(Y = \min\{n \in \{1, 2, \ldots\}:
\sum_{i=1}^n X_i = k\}\) has the negative binomial distribution with parmeters \(k\) and \(p\).
transformation
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Bernoulli distribution
Rademacher distribution
If \(X\) has the Bernoulli distribution with parameter \(\frac{1}{2}\) then \(2 X - 1\) has the Rademacher distribution.
linear transformation
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beta distribution
arcsine distribution
The beta distribution with parameters \(\alpha = \frac{1}{2}\) and \(\beta = \frac{1}{2}\) is the arcsine distribution.
special case
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beta distribution
continuous uniform distribution
The beta distribution with parameters \(\alpha = 1\) and \(\beta = 1\) is the (standard) continuous uniform distribution.
Also, the k-th order statistic from sample of n independent continuous uniform variables
is Beta(k,n+1-k).
special case
convolution
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beta general distribution
beta distribution
If \(X\) has the beta general distribution with parameters \(\alpha \in (0, \infty)\), \(\beta \in (0, \infty)\) and \(L \lt R\), then \(Y = \frac{X-L}{R-L}\) has beta distribution with parameters \(\alpha\) and \(\beta\).
transformation
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beta distribution
inverse beta distribution
If \(X\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\), then \(Y = \frac{X}{1 - X}\) has the inverse beta distribution with parameters \(\alpha\) and \(\beta\).
transformation
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beta distribution
semicircle distribution
If \(X\) has the beta distribution with parameters \(\alpha = \frac{3}{2}\) and \(\beta = \frac{3}{2}\), and \(r \in (0, \infty)\), then \(Y = r (2 X - 1)\) has the semicircle distribution with parameter \(r\).
transformation
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beta distribution
beta distribution
If \(X\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) then \(Y = 1 - X\) has the beta distribution with parameters \(\beta\) and \(\alpha\).
transformation
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beta distribution
beta distribution
If \(X\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta = 1\), and \(r \in (0, \infty)\), then \(Y = X^r\) has the beta distribution with parameters \(\frac{\alpha}{r}\) and \(1\).
transformation
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beta distribution
Pareto distribution
If \(X\) has the beta distribution with left parameter \(\alpha \in (0, \infty)\) and right parameter \(1\), then \(Y = \frac{1}{X}\) has the Pareto distribution with shape parameter \(\alpha\).
transformation
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beta distribution
binomial distribution
beta-binomial distribution
If \(P\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) and if the conditional distribution of \(X\) given \(P = p\) has the binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and \(p\), then \(X\) has the beta-binomial distribution with parameters \(n\), \(\alpha\), and \(\beta\).
Conditioning
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beta-binomial distribution
continuous uniform distribution
The standard uniform distribution is a special case of beta-binomial distribution with \( \a=\b=1 \)
special case
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beta-binomial distribution
negative hypergeometric distribution
The negative hypergeometric distribution is a special case of beta-binomial distribution with \( n=n_1, a=n_2, b=n_3 \)
special case
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binomial distribution
standard normal distribution
If \(X_n\) has the binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and fixed \(p \in (0, 1)\) then then the distribution of \(Z_n = \frac{X_n - n p}{\sqrt{n p (1 - p)}}\) converges to the standard normal distribution as \(n \to \infty\).
central limit theorem
dinov2008central
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binomial distribution
Bernoulli distribution
The binomial distribution with parameters \(n = 1\) and \(p \in [0, 1]\) is the Bernoulli distribution with parameter \(p\).
Also, if \((X_1, X_2, \ldots, X_n)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in [0, 1]\) then \(Y = \sum_{i = 1}^n X_i\) has the binomial distribution with parameters \(n\) and \(p\).
special case
convolution
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binomial distribution
binomial distribution
If \(X\) has the binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and \(p \in [0, 1]\); \(Y\) has the binomial distribution with parameters \(m \in \{1, 2, \ldots\}\) and \(p\); and \(X\) and \(Y\) are independent, then \(X + Y\) has the binomial distribution with parameters \(m + n\) and \(p\).
convolution
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binomial distribution
hypergeometric distribution
Suppose that \(\boldsymbol{X} = (X_1, X_2, \ldots)\) is a Bernoulli trials sequence with parameter \(p \in (0, 1)\). For \(n \in \{1, 2, \ldots\}\) let \(Y_n = \sum_{i=1}^n X_i\), so that \(Y_n\) has the binomial distribution with parameters \(n\) and \(p\). If \(m \lt n\) then the distribution of \(Y_m\) given \(Y_n = k\) is hypergeoemtric with parameters \(m\), \(n\), and \(k\).
Conditional distribution
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binomial distribution
Poisson distribution
The binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and \(p \in (0, 1)\) converges to the Poisson distribution with parameter \(\lambda \in (0, \infty)\) if \(n \to \infty\), \(p \to 0\), with \(n p \to \lambda\).
parameter limit
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binomial distribution
negative binomial distribution
For \(n \in \{1, 2, \ldots\}\), let \(Y_n\) denote the number of successes in the first \(n\) of a sequence of Bernoulli trials, so that \(Y_n\) has the binomial distribution with trial parameter \(n\) and sucess parameter \(p\). Then for \(k \in \{1, 2, \ldots\}\), \(Z_k = \min\{n: Y_n \geq k\} - k\) has the negative binomial distribution with stopping parameter \(k\) and success parameter \(p\).
inverse stochastic process
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Cauchy Distribution
Cauchy Distribution
If \(X\) has the Cauchy distribution with location parameter \(\alpha_1 \in (-\infty, \infty)\) and location parameter \(\beta_1 \in (0, \infty)\), \(Y\) has the Cauchy distribution with location parameter \(\alpha_2 \in (-\infty, \infty)\) and scale parameter \(\beta_2 \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the Cauchy distribution with location parameter \(\alpha_1 + \alpha_2\) and scale parameter \(\beta_1 + \beta_2\).
convolution
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Cauchy Distribution
normal Distribution
If \(Z_1\) and \(Z_2\) are two independent standard normal variables, then the ratio \(\frac{Z_1}{Z_2}\)
has the standard Cauchy distribution.
transformation
MR1326603
Cauchy Distribution
normal Distribution
If \(Z_1\) and \(Z_2\) are two independent normally distributed random
variables \(\sim N(0, \sigma^2)\), then the ratio \(\frac{Z_1}{Z_2}\)
has the standard Cauchy distribution.
transformation
MR1326603
Cauchy distribution
Cauchy distribution
If \(X\) has Cauchy distribution with location parameter \(\alpha \in (-\infty, \infty)\) and scale parameter \(\beta \in (0, \infty)\), \(a \in (-\infty, \infty)\) and \(b \in (0, \infty)\), then \(a + b X\) has the Cauchy distribution with location parameter \(a + b \alpha\) and location parameter \(\beta b\).
location-scale transformation
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chi-square distribution
chi-square distribution
If \(X\) has the chi-square distribution with \(m \in (0, \infty)\) degrees of freedom; \(Y\) has the chi-square distribution with \(n \in (0, \infty)\) degrees of freedom; and \(X\) and \(Y\) are independent, then \(X + Y\) has the chi-square distribution with \(m + n\) degrees of freedom.
convolution
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chi-square distribution
gamma distribution
If \(X\) has a chi-square distribution with \(\nu \in \{1, 2, \ldots\}\) degrees of freedom, and \(c \in (0, \infty)\) , then \(Y = c X\) has the gamma distribution with shape parameter \(k = \frac{\nu}{2}\) and scale parameter \(\theta = 2 c\).
scale transformation
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chi-square distribution
normal distribution
If \(X_n\) has the chi-square distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom, then the distribution of \(Z = \frac{X_n - n}{\sqrt{2 n}}\) converges to the standard normal distribution as \(n \to \infty\).
central limit theorem
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dinov2008central
chi-square distribution
F-distribution
If \(U\) has the chi-square distribution with \(m \in \{1, 2, \ldots\}\) degrees of freedom; \(V\) has the chi-square distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom; and \(U\) and \(V\) are independent, then \(X = \frac{U/m}{V/n}\) hs the \(F\)-distribution with \(m\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator
nonlinear transformation
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non-central chi-square distribution
chi-square distribution
If \(X\) has the non-central chi-square distribution with \(\nu \in \{1, 2, \ldots\}\) degrees of freedom and non-centrality parameter \(\lambda = 0\), then \(X\) has a chi-square distribution with \(\nu\) degrees of freedom.
special case
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chi-square distribution
chi distribution
If \(X\) has the chi-square distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom, then \(\sqrt{X}\) has the chi distribution with \(n\) degrees of freedom.
nonlinear transformation
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chi-square distribution
normal distribution
Students t distribution
If \(Z\) has the standard normal distribution, \(V\) has the chi-square distribution with \(n \in (0, \infty)\) degrees of freedom, and \(Z\) and \(V\) are independent, then \(T = \frac{Z}{\sqrt{V / n}}\) has the students's \(t\)-distribution with \(n\) degrees of freedom.
transformation
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chi-square distribution
Poisson distribution
Rice distribution
If \(X\) has the Poisson distribution with parameter \(\frac{\nu^2}{2 \sigma^2}\) where \(\nu \in (0, \infty)\) and \(\sigma \in (0, \infty)\), and the conditional distribution of \(Y\) given \(X = x \in \{0, 1, 2, \ldots\}\) is chi-square with \(2 x + 2\) degrees of freedom, then \(\sigma \sqrt{X}\) has the Rice distribution with distance parameter \(\nu\) and scale parameter \(\sigma\).
mixture and transformation
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continuous uniform distribution
continuous uniform distribution
If \(X\) is uniformly distributed on the interval \([a, b]\) and \(c, d \in (-\infty, \infty)\) with \(c \ne 0\), then \(Y = cX + d\) is uniformly distributed on \([ca + d, cb + d]\) if \(c \gt 0\) or on \([cb + d, ca + d]\) if \(c \lt 0\)
linear transformation
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continuous uniform distribution
standard uniform distribution
The continuous uniform distribution on \([0, 1]\) is the standard uniform distribution
special case
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continuous uniform distribution
triangular distribution
If \(X\) and \(Y\) are independent and each is uniformly distributed on the interval \([a, b]\), then \(X + Y\) has the triangular distribution with parameters \(a\), \(b\), and \(c = \frac{a+b}{2}\).
convolution
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continuous uniform distribution
exponential distribution
If \(X\) has the standard uniform distribution and \(\beta \in (0, \infty)\), then \(-\beta \ln(1 - X)\) has the exponential distribution with scale parameter \(\beta\).
nonlinear transformation
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continuous uniform distribution
Pareto distribution
If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\beta \in (0, \infty)\) then \(\frac{\mu}{(1 - X)^{1/\beta}}\) has the Pareto distribution with location parameter \(\mu\) and shape parameter \(\beta\).
nonlinear transformation
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continuous uniform distribution
beta distribution
If \(X\) has the standard uniform distribution and \(\alpha \in (0, \infty)\) then \(X^{1/\alpha}\) has the beta distribution with left parameter \(\alpha\) and right parameter 1.
nonlinear transformation
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continuous uniform distribution
Cauchy distribution
If \(X\) has the standard uniform distribution then \(\tan[\pi(X - \frac{1}{2})]\) has the standard Cauchy distribution.
nonlinear transformation
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continuous uniform distribution
arcsine distribution
If \(X\) has the standard uniform distribution then \(\sin^2(\frac{\pi}{2} X)\) has the arcsine distribution.
nonlinear transformation
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continuous uniform distribution
exponential-logarithmic distribution
If \(X\) has the standard uniform distribution, \(b \in (0, \infty)\), and \(p \in (0, 1)\) then \(\frac{1}{b}\ln\left(\frac{1 - p}{1 - p^{1 - X}}\right)\) has the exponential-logarithmic distribution with parameters \(b\) and \(p\).
nonlinear transformation
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continuous uniform distribution
geometric distribution
If \(X\) has the standard uniform distribution and \(p \in (0, 1)\) then \(\lceil \frac{\ln(1 - X)}{\ln(1 - p)}\rceil\) has the geometric distribution with parameter \(p\).
nonlinear transformation
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continuous uniform distribution
Gumbel distribution
If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\sigma \in (0, \infty)\) then \(\mu - \sigma \ln(-\ln(X))\) has the Gumbel distribution with location prarameter \(\mu\) and scale parameter \(\sigma\).
nonlinear transformation
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continuous uniform distribution
hyperbolic secant distribution
If \(X\) has the standard uniform distribution then \(\frac{2}{\pi} \ln[\tan(\frac{\pi}{2} X)]\) has the hyperbolic secant distribution.
nonlinear transformation
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continuous uniform distribution
Laplace distribution
If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), \(b \in (0, \infty)\), then \(\mu + b \ln(2 \min\{X, 1 - X\})\) has the Laplace distribution with location parameter \(\mu\) and scale parameter \(b\).
nonlinear transformation
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continuous uniform distribution
logistic distribution
If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\sigma \in (0, \infty)\), then \(\mu + \sigma \ln\left(\frac{X}{1 - X}\right)\) has the logistic distribution with location parameter \(\mu\) and scale parameter \(\sigma\).
nonlinear transformation
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continuous uniform distribution
log-logistic distribution
If \(X\) has the standard uniform distribution, \(\alpha \in (0, \infty)\), and \(\beta \in (0, \infty)\), then \(\alpha \left(\frac{X}{1 - X}\right)^{1/\beta}\) has the log-logistic distribution with scale parameter \(\alpha\) and shape parameter \(\beta\).
nonlinear transformation
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continuous uniform distribution
Rayleigh distribution
If \(X\) has the standard uniform distribution and \(\sigma \in (0, \infty)\), then \(\sigma \sqrt{-2 \ln(1 - X)}\) has the Rayleigh distribution with scale parameter \(\sigma\).
nonlinear transformation
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continuous uniform distribution
Weibull distribution
If \(X\) has the standard uniform distribution, \(\sigma \in (0, \infty)\) and \(\alpha \in (0, \infty)\), then \(\sigma (-\ln(1 - X))^{1/\alpha}\) has the Weibull distribution with shape parameter \(\alpha\) and scale parameter \(\sigma\).
nonlinear transformation
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continuous uniform distribution
Irwin-Hall distribution
If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the standard uniform distribution, then \(sum_{i=1}^n X_i\) has the Irwin-Hall distribution with parameter \(n\).
convolution
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exponential distribution
exponential distribution
If \(X\) has the exponential distribution with rate parameter \(r \in (0, \infty)\), \(Y\) has the exponential distribution with rate parameter \(s \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(\min\{X, Y\}\) has the exponential distribution with rate parameter \(r + s\).
nonlinear transformation
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exponential distribution
continuous uniform distribution
If \(X\) has the exponential distribution with parameter \(\lambda \in (0, \infty)\) then \(Y = e^{-\lambda X}\) has the standard uniform distribution.
transformation
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exponential distribution
gamma distribution
If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the exponential distribution with parameter \(\lambda \in (0, \infty)\) then \(Y = \sum_{i=1}^n X_i\) has the gamma distribution with shape parameter \(n\) and scale parameter \(\frac{1}{\lambda}\).
Also, If \(X\) has the gamma distributed with parameter shape parameter \(k = 1\) and scale parameter \(\lambda \in (0, \infty)\) then and then \(X\) has the exponential distribution with scale parameter \(\lambda\) (and hence rate parameter \(1/\lambda\).
convolution
special case
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exponential distribution
Pareto distribution
If \(a \in (0, \infty)\) and \(X\) has the exponential distribution with parameter \(\lambda \in (0, \infty)\) then \(Y = a e^X\) has the Pareto distribution with scale parameter \(a\) and shape parameter \(\lambda\).
nonlinear transformation
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exponential distribution
Weibull distribution
If \(X\) has the standard exponential distribution, \(k \in (0, \infty)\), and \(b \in (0, \infty)\), then \(Y = b X^{1/k}\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b\).
nonlinear transformation
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exponential distribution
extreme value distribution
If \((X_1, X_2, \ldots)\) is a sequence of indpendent random variables, each with the standard exponential distribution, then the distribution of \(\max\{X_1, \ldots, X_n\} - \ln(n)\) converges to the standard Gumbel distribution.
limiting distribution
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exponential distribution
Laplace distribution
If \(X\) and \(Y\) are independent random variables and each has the exponential distribution with scale parameter \(\sigma \in (0, \infty)\) then \(X - Y\) has the Laplace distribution with location parameter \(0\) and scale parameter \(\sigma\).
convolution
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exponential distribution
Rademacher distribution
Laplace distribution
If \(X\) has the exponential distribution with scale parameter \(\sigma \in (0, \infty)\), \(Y\) has the Rademacher distribution, and \(X\) and \(Y\) are independent, then \(X V\) has the Laplace distribution with location parameter \(0\) and scale parameter \(\sigma\).
nonlinear transformation
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exponential distribution
normal distribution
Laplace distribution
If \(X\) has the standard exponential distribution, \(Z\) has the standard normal distribution, \(X\) and \(Z\) are independent, \(\mu \in (-\infty, \infty)\), and \(\sigma \in (0, \infty)\), then \(\mu + \sigma Z \sqrt{2 X}\) has the Laplace distribution with location parameter \(\mu\) and scale parameter \(\sigma\).
nonlinear transformation
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F-distribution
F-distribution
If \(X\) has the F-distribution with \(m \in \{1, 2, \ldots\}\) degrees of freedom in the numerator and \(n \in \{1, 2, \ldots\}\) degrees of freedom in the denominator, then \(\frac{1}{X}\) has the F-distribution with \(n\) degrees of freedom in the numerator and \(m\) degrees of freedom in the denominator.
nonlinear transformation
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F-distribution
beta distribution
If \(X\) has the F-distribution with \(m \in (0, \infty)\) degrees of freedom in the numerator and \(n \in (0, \infty)\) degrees of freedom in the denominator, then \(\frac{(m/n)X}{1 + (m/n)X}\) has the beta distribution with left parameter \(\frac{m}{2}\) and right parameter \(\frac{n}{2}\).
nonlinear transformation
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F-distribution
chi-square distribution
If \(X\) has the chi-square distribution with \(m \in \{1, 2, \ldots\}\) degrees of freedom in the numerator and \(n \in \{1, 2, \ldots\}\) degrees of freedom in the denominator, then the distribution of \(m X\) converges to the chi-square distribution with \(m\) degrees of freedom as \(n \to \infty\).
limiting distribution
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gamma distribution
gamma distribution
If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\) and \(c \in (0, \infty)\), then \(Y = cX\) has the gamma distribution with shape parameter \(\alpha\) and scale parameter \(c \lambda\).
scale transformation
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gamma distribution
gamma distribution
If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), \(Y\) has the gamma distribution with shape parameter \(\beta \in (0, \infty)\) and scale parameter \(\lambda\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the gamma distribution with shape parameter \(\alpha + \beta\) and scale parameter \(\lambda\).
convolution
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gamma distribution
exponential distribution
If \(X\) has the gamma distributed with parameter shape parameter \(k = 1\) and scale parameter \(\lambda \in (0, \infty)\) then and then \(X\) has the exponential distribution with scale parameter \(\lambda\) (and hence rate parameter \(1/\lambda\).
special case
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gamma distribution
chi-square distribution
If \(X\) has the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), then \(\frac{2 X}{\lambda}\) has the chi-square distribution with \(k\) degrees of freedom.
linear transformation
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gamma distribution
Erlang distribution
If \(X\) has the gamma distribution with shape parameter \(k \in \{1, 2, \ldots\}\) and scale parameter \(c \in (0, \infty)\), then \(X\) has the Erlang distribution with shape parameter \(k\) and scale parameter \(c\).
special case
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gamma distribution
Maxwell-Boltzmann distribution
If \(X\) has the gamma distribution with shape parameter \(k = \frac{3}{2}\) and scale parameter \(\theta = 2 a^2\) where \(a \in (0, \infty)\), then \(\sqrt{X}\) has the Maxwell-Boltzmann distribution with parameter \(a\).
nonlinear transformation
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gamma distribution
normal distribution
If \(X_k\) has the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) then the distribution of \(\frac{X - k b}{\sqrt{k} b}\) converges to the standard normal distribution as \(k \to \infty\).
central limit theorem
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gamma distribution
beta distribution
If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), \(Y\) has the gamma distribution with shape parameter \(\beta \in (0, \infty)\) and scale parameter \(\lambda\), and \(X\) and \(Y\) are independent, then \(\frac{X}{X + Y}\) has the beta distribution with left parameter \(\alpha\) and right parameter \(\beta\).
nonlinear transformation
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gamma distribution
inverted beta distribution
If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), \(Y\) has the gamma distribution with shape parameter \(\beta \in (0, \infty)\) and scale parameter \(\lambda\), and \(X\) and \(Y\) are independent, then \(\frac{X}{Y}\) has the inverted beta distribution with shape parameters \(\alpha\) and \(\beta\).
nonlinear transformation
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gamma distribution
Levy distribution
If \(X\) has the gamma distribution with shape parameter \(\frac{1}{x}\) and scale parameter \(\sigma \in (0, \infty)\) then \(\frac{1}{X}\) has the Levy distribution with location parameter \(0\) and scale parameter \(\frac{2}{\sigma}\).
nonlinear transformation
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geometric distribution
geometric distribution
If \(X\) has the geometric distributin on \(\{0, 1, \ldots\}\) then \(X + 1\) has the geometric distribution on \(\{1, 2 \ldots\}\).
linear transformation
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geometric distribution
discrete uniform distribution
If \(X\) has the geometric distribution on \(\{1, 2, \ldots\}\) with parameter \(p \in (0, 1)\) and \(n \in \{1, 2, \ldots\}\), then the conditional distribution of \(X\) given \(X \in \{1, 2, \ldots, n\}\) converges to the uniform distribution on \(\{1, 2, \ldots, n\}\) as \(p \to 0\).
limiting conditional distribution
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geometric distribution
exponential distribution
If \(X_n\) has the geometric distribution on \(\{1, 2, \ldots\}\) with parmeter \(p_n \in (0, 1)\) for each \(n \in \{1, 2, \ldots\}\) and \(n p_n \to r \in (0, \infty)\) as \(n \to \infty\), then the distribution of \(\frac{X_n}{n}\) converges to the exponential distribution with rate parameter \(r\).
limiting distribution
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dinov2008central
extreme value distribution
extreme value distribution
If \(X\) has the Gumbel distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), and \(a \in (-\infty, \infty)\), \(b \in (0, \infty)\), then \(a + b X\) has the Gumbel distribution with location parameter \(a + b \mu\) and scale parameter \(b \sigma\).
location-scale transformation
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Gumbel distribution
standard uniform distribution
If \(X\) has the Gumbel distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\) then \(\exp\left[-\exp\left(\frac{X - \mu}{\sigma}\right)\right]\) has the standard uniform distribution.
nonlinear transformation
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hypergeometric distribution
hypergeometric distribution
If \(X\) has the hypergeometric distribution with population size \(m \in \{1, 2, \ldots\}\), sample size \(n \in \{1, 2, \ldots, m\}\) and type parameter \(r \in \{1, 2, \ldots, m\}\) then \(n - X\) has the hypergeometric distribution with population size \(m\), sample size \(n\), and type parameter \(m - r\).
linear transformation
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hypergeometric distribution
binomial distribution
Let \(n \in \{1, 2, \ldots\}\) and \(r_m \in \{1, 2, \ldots, m\}\) for each \(m \in \{1, 2, \ldots\}\) with \(\frac{r_m}{m} \to p \in (0, 1)\) as \(m \to \infty\). The hypergeometric distribution with population size \(m\), sample size \(n\), and type parameter \(r_m\) converges to the binomial distribution with trial parameter \(n\) and success parameter \(p\) as \(m \to \infty\).
limiting distribution
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hypergeometric distribution
Bernoulli distribution
If \(X\) has the hypergeometric distribution with population size \(m \in \{1, 2, \ldots\}\), sample size \(n = 1\), and type parameter \(r \in \{1, 2, \ldots, m\}\), then \(X\) has the Bernoulli distribution with parameter \(\frac{r}{m}\).
TBD
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hyperbolic secant distribution
continuous uniform distribution
If \(X\) has the hyperbolic secant distribution then \(\frac{2}{\pi} \arctan[\exp(\frac{\pi}{2} X)]\) has the standard uniform distribution.
nonlinear transformation
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Irwin-Hall distribution
Irwin-Hall distribution
If \(X\) has the Irwin-Hall distribution with parameter \(m \in \{1, 2, \ldots\}\), \(Y\) has the Irwin-Hall distribution with parameter \(n \in \{1, 2, \ldots\}\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the Irwin-Hall distribution with parameter \(m + n\).
convolution
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Irwin-Hall distribution
standard uniform distribution
The Irwin-Hall distribution with parameter \(1\) is the standard uniform distribution.
special case
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Irwin-Hall distribution
triangular distribution
The Irwin-Hall distribution with parmeter \(2\) is the triangular distribution with left endpoint \(0\), right endpoint \(1\) and midpoint \(\frac{1}{2}\).
special case
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inverted beta distribution
inverted beta distribution
If \(X\) has the inverted beta distribution with shape parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) then \(\frac{1}{X}\) has the inverted beta distribution with shape parameters \(\beta\) and \(\alpha\).
nonlinear transformation
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inverted beta distribution
F-distribution
If \(X\) has the inverted beta distribution with shape parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) then \(\frac{\beta}{\alpha} X\) has the F-distribution with \(2 \alpha\) degrees of freedom in the numerator and \(2 \beta\) degrees of freedom in the denominator.
linear transformation
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Laplace distribution
exponential distribution
If \(X\) has the Laplace distribution with location parameter \(0\) and scale parameter \(\sigma \in (0, \infty)\) then \(|X|\) has the exponential distribution with scale parameter \(\sigma\).
nonlinear transformation
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Levy distribution
folded normal distribution
If \(X\) has the Levy distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), then \(\frac{1}{\sqrt{X - \mu}}\) has the folded normal distribution with location parameter \(0\) and scale parameter \(\frac{1}{\sigma}\).
nonlinear transformation
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Levy distribution
gamma distribution
If \(X\) has the Levy distribution with location parameter \(0\) and scale parameter \(\sigma \in (0, \infty)\), then \(\frac{1}{X}\) has the gamma distribution with shape parameter \(\frac{1}{x}\) and scale parameter \(\frac{2}{\sigma}\).
nonlinear transformation
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logarithmic distribution
Poisson distribution
negative binomial distribution
If \((X_1, X_2, \ldots)\) is a sequence of independent random variables, each with the logarithmic distribution with parameter \(p \in (0, 1)\) and \(N\) has the Poisson distribution with parameter \(\lambda \in (0, \infty)\), then \(\sum_{i=1}^N X_i\) has the negative binomial distribution with parameters \(\lambda\) and \(p\).
mixture
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%logistic to standard uniform
logistic distribution
continuous uniform distribution
If \(X\) has the logistic distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\) then \(\frac{1}{1 + \exp\left(\frac{X - \mu}{\sigma}\right)}\) has the standard uniform distribution.
nonlinear transformation
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skew logistic distribution
exponential distribution
If \(X\) has the skew-logistic distribution with parameter \(\alpha \in (0, \infty)\), then \(Y = \ln(1+e^{-X})\) has the exponential distribution with rate parameter \(\alpha\).
transformation
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log-normal distribution
log-normal distribution
If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), \(Y\) has the log-normal distribuiton with location parameter \(\nu \in (-\infty, \infty)\) and scale parameter \(\tau \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(X Y\) has the log-normal distirbution with location parameter \(\mu + \tau\) and scale parameter \(\sqrt{\sigma^2 + \tau^2}\).
nonlinear transformation
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log-normal distribution
log-normal distribution
If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parmaeter \(\sigma \in (0, \infty)\), and \(a \neq 0\) then \(a X\) has the log-normal distribution with location parameter \(a \mu\) and scale parameter \(|a| \sigma\).
nonlinear transformation
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log-normal distribution
log-normal distribution
If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(a \in (0, \infty)\), and \(\sigma \in (0, \infty)\), then \(a X\) has the log-normal distribution with location parameter \(\ln(a) + \mu\) and scale parameter \(\sigma\).
linear transformation
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log-normal distribution
normal distribution
If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), then \(\ln(X)\) has the normald distribution with location parameter \(\mu\) and scale parameter \(\sigma\).
nonlinear transformation
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Maxwell-Boltzmann distribution
Maxwell-Boltzmann distribution
If \(X\) has the Maxwell-Boltzmann distribution with scale parameter \(a \in (0, \infty)\) and \(b \in (0, \infty)\), then \(b X\) has the Maxwell-Boltzmann distribution with scale parameter \(a b\).
scale transformation
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Maxwell-Boltzmann distribution
chi distribution
If \(X\) has the Maxwell-Boltzmann distribution with scale parameter \(a \in (0, \infty)\), then \(\frac{X}{a}\) has the chi distribution with 3 degrees of freedom.
scale transformation
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negative binomial distribution
negative binomial distribution
If \(X\) has the negative binomial distribution with stopping parameter \(r \in (0, \infty)\) and success parameter \(p \in (0, 1)\), \(Y\) has the negative binomial distribution with stopping parameter \(s \in (0, \infty)\) and success parameter \(p\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the negative binomial distribution with stopping parameter \(r + s\) and success parameter \(p\).
convolution
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negative binomial distribution
geometric distribution
The negative binomial distribution with stopping parameter \(1\) and success parameter \(p \in (0, 1)\) is the geometric distribution with success parameter \(p\).
Also, if \((X_1, X_2, \ldots, X_n)\) is a sequence of independent Geometric(p) random variables, their sum \(\sum_{i=1}^n{X_i} is negative binomial(n,p).
special case
convolution
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negative binomial distribution
Poisson distribution
If \(p_r \in (0, 1)\) for each \(r \in (0, \infty)\) and \(r \frac{p}{1-p} \to \lambda \in (0, \infty)\) as \(r \to \infty\), then the negative binomial distribution with stopping parameter \(r\) and success parameter \(p_r\) converges to the Poisson distribution with parameter \(\lambda\).
limiting distribution with respect to parameter
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Poisson distribution
Exponential distribution
If for every t > 0 the number of arrivals in the time interval [0,t] follows
the Poisson distribution with mean \(\lambda t\), then the sequence of inter-arrival
times are independent and identically distributed exponential random variables
with mean \(\fract{1}{λ}\).
special case
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negative binomial distribution
normal distribution
If \(X\) has the negative binomial distribution with stopping parameter \(r \in (0, \infty)\) and success parameter \(p \in (0, \infty)\), then the distribution of \(\frac{p X - r (1 - p)}{\sqrt{r (1 - p}}\) converges to the standard normal distribution at \(r \to \infty\).
central limit theorem
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negative binomial distribution
binomial distribution
For \(k \in \{1, 2, \ldots\}\), let \(Z_k\) denote the number of failures before the \(k\)th success in a sequence of Bernoulli trials with success parameter \(p \in (0, 1)\), so that \(Z_k\) has the negative binomial distribution with stopping parameter \(k\) and success parameter \(p\). Then for \(n \in \{1, 2, \ldots\}\), \(Y_n = \max\{k: k + Z_k \leq n\}\) has the binomial distribution with trial parameter \(n\) and success parameter \(p\).
inverse stochastic process
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normal distribution
log-normal distribution
If \(X\) has a normal distribution with mean \(\mu \in (-\infty, \infty)\) and variance \(\sigma^2\), then \(Y = e^X\) has the log-normal distribution with parameters \(\mu\) and \(\sigma^2\).
transformation
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normal distribution
folded normal Distribution
If \(X\) is has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then \(|X|\) has the folded normal distribution with parameters \(\mu\) and \(\sigma\).
transformation
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normal distribution
half normal distribution
If \(X\) is has the normal distribution with mean \(\mu\) = 0 and standard deviation \(\sigma \in (0, \infty)\), then \(|X|\) has a half-normal distribution with parameter \(\sigma\).
special case
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normal distribution
non-central chi-square distribution
If \(X\) has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then variable \(Y = \frac{X^2}{\sigma^2}\) has a non-central chi-square distribution with one degree of freedom and non-centrality parameter \(\frac{\mu^2}{\sigma^2}\).
transformation
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normal distribution
truncated normal distribution
If \(X\) is has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), and if \(a, b \in [-\infty, \infty]\) with \(a \lt b\), then the conditional distribution of \(X\) given \(X \in (a,b)\) is the truncated normal distribution with location parameter \(\mu\), scale parameter \(\sigma\), minimum value \(a\), and maximum value \(b\).
conditioning
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normal distribution
Levy distribution
If \(X\) has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then \(\frac{1}{(X - \mu)^2}\) has the Levy distribution with location parameter 0 and scale parameter \(\frac{1}{\sigma^2}\).
transformation
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normal distribution
Rice distribution
Let \(\nu \in [0, \infty)\), \(\theta \in (-\infty, \infty)\) and \(\sigma \in (0, \infty)\). If \(X\) has the normal distribution with mean \(\nu \cos(\theta)\) and standard deviation \(\sigma\), \(Y\) has the normal distribution with mean \(\nu \sin(\theta)\) and standard deviation \(\sigma\), and \(X\) and \(Y\) are independent, then \(\sqrt{X^2 + Y^2}\) has the Rice distribution with distance parameter \(\nu\) and scale parameter \(\sigma\).
nonlinear transformation
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normal distribution
normal distribution
If \(X\) has the normal distribution with mean \(\mu = 0\) and standard deviation \(\sigma = 1\), then \(X\) has a standard normal distribution.
special case
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normal distribution
chi-square distribution
If \(X_1, X_2, \ldots, X_n\) are independent standard normal random variables, then \(\sum_{i=1}^n X_i^2\) has the chi-square distribution with \(n\) degrees of freedom.
convolution
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normal distribution
Students t distribution
If \(X_1, X_2, \ldots, X_n\) are independent normally distributed random variables with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then \(T = \frac{\overline{X} - \mu}{S / \sqrt{n}}\) has the students t distribution with \(n-1\) degrees of freedom.
transformation
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normal distribution
Maxwell-Boltzmann distribution
If \(X_1\), \(X_2\), and \(X_3\) are independent random variables, each with the normal distribuiton with mean \(0\) and standard deviation \(a \in (0, \infty)\), then \(\sqrt{X_1^2 + X_2^2 + X_3^2}\) has the Maxwell-Boltzmann distribution with parameter \(a\).
nonlinear transformation
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normal distribution
Cauchy distribution
If \(X\) and \(Y\) are independent variables, each with the standard normal distribution, then \(\frac{X}{Y}\) has the standard Cauchy distribution.
nonlinear transformation
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Pareto distribution
exponential distribution
If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), then \(\ln\left(\frac{X}{b}\right)\) has the exponential distribution with rate parameter \(a\).
nonlinear transformation
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Pareto distribution
Pareto distribution
If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), and \(c \in (0, \infty)\) then \(c X\) has the Pareto distribution with shape parameter \(a\) and scale parameter \(b c\).
scale transformation
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Pareto distribution
beta distribution
If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) then \(\frac{b}{X}\) has the beta distribution with left shape parameter \(a\) and right shape parameter \(1\).
nonlinear transformation
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Pareto distribution
continuous uniform distribution
If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), then \(1 - \left(\frac{b}{X}\right)^a\) has the standard uniform distribution.
nonlinear transformation
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Poisson distribution
Poisson distribution
If \(X\) has the Poisson distribution with parameter \(\alpha \in (0, \infty)\), \(Y\) has the Poisson distribution with parameter \(\beta \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the Poisson distribution with parameter \(\alpha + \beta\).
convolution
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Poisson distribution
normal distribution
If \(X\) has the Poisson distribution with parameter \(\alpha \in (0, \infty)\), then the distribution of \(\frac{X - \alpha}{\sqrt{\alpha}}\) converges to the standard normal distribution as \(\alpha \to \infty\).
central limit theorem
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Poisson distribution
normal distribution
As \( \sigma^2=\mu and \mu\to\infty \) Poisson distribution becomes normal distribution.
limiting
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Poisson distribution
binomial distribution
If \(\{N_t: t \ge 0\}\) is a Poisson process and if \(s \lt t\), then the conditional distribution of \(N_s\) given \(N_t = n\) is binomial with parameters \(n\) and \(\frac{s}{t}\).
conditioning
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Poisson distribution
gamma distribution
If \(\{N_t: t \ge 0\}\) is a Poisson process with rate parameter \(\alpha \in (0, \infty)\) and \(n \in \{1, 2, \ldots\}\) then \(T = \min\{t \ge 0: N_t = n\}\) has the gamma distsribution with shape parameter \(k\) and scale parameter \(\frac{1}{\alpha}\).
stochastic process
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Poisson distribution
logarithmic distribution
negative binomial distribution
If \(\bs{X} =(X_1, X_2, \ldots)\) is a sequence of independent random variables, each with the logarithmic distribution with parameter \(p \in (0, 1)\), \(N\) has the Poisson distribution with parameter \(-r \ln(1 - p)\) where \(r \in (0, \infty)\), and \(N\) and \(\bs{X}\) are independent, then \(\sum_{i=1}^N X_i\) has the negative binomial distribution with stopping parameter \(r\) and sucess parameter \(p\).
compound Poisson transformation
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Poisson distribution
gamma distribution
negative binomial distribution
If \(\Lambda\) has the gamma distribution with shape parameter \(r \in (0, \infty)\) and scale parameter \(\frac{p}{1-p}\) where \(p \in (0, 1)\), and the conditional distribution of \(X\) given \(\Lambda = \lambda \in (0, \infty)\) is Poisson with parameter \(\lambda\), then \(X\) has the negative binomial distribution with stopping parameter \(r\) and success parameter \(p\).
mixture
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Rademacher distribution
Bernoulli distribution
If \(X\) has the Rademacher distribution then \(\frac{X+1}{2}\) has the Bernoulli distribution with success parameter \(\frac{1}{2}\).
linear transformation
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Rayleigh distribution
chi-square distribution
If \(X\) has the Rayleigh distribution with scale parameter \(1\), then \(X^2\) has the chi-square distribution with 2 degrees of freedom.
nonlinear transformation
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Rayleigh distribution
Rayleigh distribution
If \(X\) has the Rayleigh distribution with scale parameter \(\sigma \in (0, \infty)\) and \(b \in (0, \infty)\), then \(b X\) has the Rayleigh distribution with scale parameter \(b \sigma\).
scale transformation
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Rayleigh distribution
gamma distribution
If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the Rayleigh distribution with scale parameter \(\sigma \in (0, \infty)\), then \(\sum_{i=1}^n X_i^2\) has the chi-square distribution with shape parameter \(n\) and scale parameter \(2 \sigma^2\).
nonlinear transformation
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Rayleigh distribution
continuous uniform distribution
If \(X\) has the Rayleigh distribution with shape parameter \(\sigma \in (0, \infty)\), then \(1 - \exp\left(-\frac{X^2}{2 \sigma^2}\right)\) has the standard uniform distribution.
nonlinear transformation
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Rice distribution
Rayleigh distribution
The Rice distribution with distance parameter \(0\) and scale parameter \(\sigma \in (0, \infty)\) is the Rayleigh distribution with scale parameter \(\sigma\)
special case
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Rice distribution
noncentral chi-square distribution
If \(X\) has the Rice distribution with distance parameter \(\nu \in [0, \infty)\) and scale parameter \(1\), then \(X^2\) has the noncentral chi-square distribution with 2 degrees of freedom and noncentrality parameter \(\nu^2\).
nonlinear transformation
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semicircle distribution
continuous uniform distribution
If \(X\) has the semicircle distribution with radius \(r \in (0, \infty)\) then \(\frac{1}{2} + \frac{1}{\pi r^2} X \sqrt{r^2 - X^2} + \frac{1}{\pi} \arcsin\left(\frac{X}{r}\right)\) has the standard uniform distribution
nonlinear transformation
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stable distribution
Cauchy distribution
If \(X\) has a stable distribution with stability parameter \(\alpha = 1\), skewness parameter \(\beta = 0\), location parameter \(\mu \in (-\infty, \infty)\), and scale parameter \(\gamma \in (0, \infty)\), then \(X\) has a Cauchy distribution with scale parameter \(\gamma\) and location parameter \(\mu\).
special case.
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stable Distribution
normal Distribution
If \(X\) has a stable distribution with stability parameter \(\alpha = 2\), location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\gamma \in (0, \infty)\), then \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma^2 = 2 \gamma^2\).
special case.
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stable Distribution
Levy Distribution
If \(X\) has a stable distribution with stability parameter \(\alpha = \frac{1}{2}\), skewness parameter \(\beta=1\), location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\gamma \in (0, \infty)\), then \(X\) has a Levy distribution with scale parameter \(\gamma\) and shift parameter \(\mu\).
special case.
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stable Distribution
Landau Distribution
If \(X\) has a stable distribution with stability parameter \(\alpha = 1\), skewness parameter \(\beta = 1\), location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\gamma \in (0, \infty)\) then \(X\) has a Landau distribution with scale parameter \(\gamma\) and location parameter \(\mu\).
special case
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Students t distribution
F distribution
if \(X\) has the students t-distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom, then \(Y = X^2\) has the F distribuiton with \(1\) degree of freedom in the numerator and \(n\) degrees of freedom in the denominator.
transformation
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Students t distribution
Cauchy distribution
The students t-distribution with 1 degree of freedom is the standard Cauchy distribuiton.
special case
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U-quadratic distribution
continuous uniform distribuiton
If \(X\) has the U-quadratic distribution with left endpoint \(a \in (-\infty, \infty)\) and right endpoint \(b \in (a, \infty)\) then \(\frac{\alpha}{3} [(X - \beta)^3 + (\beta - \alpha)^3]\) has the standard uniform distribution, where \(\alpha = \frac{12}{(b - a)^3}\) and \(\beta = \frac{a + b}{2}\).
nonlinear transformation
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von Mises distribution
continuous uniform distribution
The von Mises distribution with location parameter \(0\) and shape parameter \(0\) is the uniform distribution on the interval \([-\pi, \pi]\).
special case
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Wald distribution
Wald distribution
If \(X\) has the Wald distribution with mean \(\mu \in (0, \infty)\) and shape parameter \(\lambda \in (0, \infty)\) and \(t \in (0, \infty)\), then \(t X\) has the Wald distribution with mean \(t \mu\) and shape parameter \(t \lambda\)
scale transformation
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Wald distribution
Wald distribution
If \(X\) has the Wald distribution with mean \(\mu a\) and shape paramter \(\lambda a^2\) where \(\mu \in (0, \infty)\), \(\lambda \in (0, \infty)\), and \(a \in (0, \infty)\), and if \(Y\) has the Wald distribution with mean \(\mu b\) and shape parameter \(\lambda b\) where \(b \in (0, \infty)\), and if \(X\) and \(Y\) are independent, then \(X + Y\) has the Wald distribution with mean \(\mu(a + b)\) and shape paramter \(\lambda(a^2 + b^2)\).
convolution
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Weibull distribution
Weibull distribution
If \(X\) has the Weibull distribution with shape parameter \(k \in (0, \infty)\), scale parameter \(b \in (0, \infty)\), and \(c \in (0, \infty)\), then \(Y = c X\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b c\).
scale transformation
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Weibull distribution
exponential distribution
If \(X\) has the Weibull distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), then \(Y = \left(\frac{X}{b}\right)^k\) has the standard exponential distribution.
transformation
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normal distribution
normal distribution
The normal distribution with \(\mu=0\) and \(\sigma^2=1\) is called the standard normal
transformation
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student distribution
normal distribution
As \(n\longrightarrow\infty\), the t-distribution approaches the normal distribution with mean 0 and variance 1
limiting
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F distribution
student distribution
The square root of a Fisher's F distribution is a students t distribution
transformation
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binomial distribution
normal distribution
If n is large enough, then the skew of the distribution is not too great. In this case, if a suitable continuity correction is used, then an excellent approximation to \(B(n, p)\) is given by the normal distribution \(N(np, np(1-p))\) as \(n \rightarrow \infty\)
limiting
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Erlang distribution
chi-square distribution
When the scale parameter \(\mu\) equals 2, then the Erlang distribution simplifies to the chi-square distribution with \(2k\) degrees of freedom
special case
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noncentral student t distribution
normal distribution
If \(T\) is noncentral t-distributed with \(\nu\) degrees of freedom and noncentrality parameter \(\mu\) and \(Z=\lim_{\nu\to\infty}T\), then \(Z\) has a normal distribution with mean \(\mu\) and unit variance
limiting
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continuous uniform distribution
Pareto distribution
If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\beta \in (0, \infty)\) then \(\frac{\mu}{(1 - X)^{1/\beta}}\) has the Pareto distribution with location parameter \(\mu\) and shape parameter \(\beta\).
nonlinear transformation
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continuous uniform distribution
exponential distribution
If \(X\) has the standard uniform distribution and \(\beta \in (0, \infty)\), then \(-\beta \ln(1 - X)\) has the exponential distribution with scale parameter \(\beta\).
nonlinear transformation
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Zipf distribution
discrete uniform distribution
The discrete uniform distribution is a special case of the Zipf distribution where \(a=0, a=1, b=n\)
special case
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Fisher-Tippett distribution
Gumbel distribution
The Gumbel distribution is a particular case of the Fisher-Tippett distribution where \(\mu=0, \beta=1\)
special case
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log-normal distribution
Gibrat's distribution
Gibrat's law is a special case of the log-normal distribution where
special case \(\mu=0, x=1\)
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Cauchy distribution
Cauchy distribution
If \(X\) is a standard Cauchy distribution, then \(Y = x_0 + \gamma X\) is a Cauchy distribution
transformation
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multinomial distribution
Binomial distribution
When \(k=2\), the multinomial distribution is the binomial distribution
transformation
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power series distribution
Pascal distribution
The power series\(c, (A(c))\) distribution becomes a Pascal distribution when \(A(c)=(1-c)^{-x}, c=1-p\)
transformation
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power series distribution
logarithmic distribution
The power series distribution is a special case of Power series distribution distribution with \(A(c)=-\log (1-c)\)
special case
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Poisson distribution
gamma-Poisson distribution
Let \(\mu \sim Gamma(\alpha, \beta)\) denote that \(\mu\) is distributed
according to the Gamma density g parameterized in terms of a shape parameter
\(\alpha\) and an inverse scale parameter \(\beta\): \(g(\mu \mid \alpha, \beta) =
\frac{\beta^{\alpha}}{\Gamma(\alpha)} \mu^{\alpha-1} e^{-\beta \mu}, \mu>0\).
Then, given the same sample of n measured values \(k_i\) as before, and a prior of
\(Gamma(\alpha, \beta)\), the posterior distribution is
\(\mu \sim Gamma(\alpha + \sum_{i=1}^n k_i, \beta + n)\).
The posterior predictive distribution of additional data is a Gamma-Poisson
distribution.
Bayesian
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gamma-Poisson distribution
Pascal (negative-binomial) distribution
The gamma-Poisson distribution becomes a Pascal (negative-binomial) distribution when \( \alpha=\frac{1-p}{p}, \beta=n \).
The negative binomial distribution arises as a continuous mixture of Poisson distributions
where the mixing distribution of the Poisson rate is a gamma distribution.
Thus, negative binomial is equivalent to a Poisson(λ) distribution, where λ is a gamma distributed
random variable, with shape = r and scale \(θ = \frac{p}{1 − p}\) or correspondingly rate
\(β = \frac{1 - p}{p}\).
transformation
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continuous uniform distribution
beta-binomial distribution
For \(a = b = 1\), the continuous uniform distribution reduces to the beta-binomial distribution as a special case
special case
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Zipf distribution
zeta distribution
The zeta distribution is equivalent to the Zipf distribution for infinite N.
limiting
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power series distribution
Poisson distribution
The power series\((c, A(c))\) distribution becomes a Poisson distribution when \(\mu=c, A(c)=e^c\)
transformation
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Pascal distribution
Poisson distribution
Consider a sequence of negative binomial distributions where the stopping
parameter n goes to infinity, whereas the probability of success in each trial,
p, goes to zero in such a way as to keep the mean of the distribution constant.
Denoting this mean \(\mu\), the parameter p will have to be
\(\mu = r \frac{p}{1-p} \rightarrow p = \frac{\mu}{r+\mu}\). Then \(Poisson(\mu) =
\lim_{n \to \infty} Pascal(n, \frac{\mu}{\mu+n})\).
transformation, limiting
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negative hypergeometric distribution
binomial distribution
As \(n_3\to\infty, n_1\to\infty\) and letting \(p=n_1/n_3, n_2=n\), the negative hypergeometric\((n_1, n_2, n_3)\) distribution becomes a binomial\((n, p)\) distribution
transformation, limiting
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negative hypergeometric distribution
Pascal distribution
As \(N\to\infty, \frac{K}{N}\to\infty\), the negative hypergeometric\((n_1, n_2, n_3)\) distribution becomes a Pascal\((n, p)\) distribution
limiting
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negative hypergeometric distribution
negative binomial distribution
As \(N\to\infty, \frac{K}{N}\to\infty\), the negative hypergeometric\((n_1, n_2, n_3)\) distribution becomes a negative binomial\((n, p)\) distribution
limiting
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Pascal distribution
geometric distribution
The geometric distribution is a special case of the Pascal distribution where \(n=1\)
special case
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discrete Weibull distribution
geometric distribution
The geometric distribution is a particular case of the discrete Weibull distribution where \(\beta=1\)
special case
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normal distribution
chi-square distribution
If \(X_i \sim Normal(\mu, \sigma^2)\), with \(i=1,...,k\) independent
random variables, then \(\sum_{i=1}^{k} (\frac{X_i-\mu}{\sigma})^2\) is a chi-sqaure
distribution.
transformation
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normal distribution
gamma-normal distribution
When the \(\sigma\) in the normal distribution is Inverted gamma\((\alpha, \beta)\), the normal distribution becomes a gamma-normal\(\mu, \alpha, \beta)\) distribution
Bayesian
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Wald distribution
normal distribution
As \(\lambda \to \infty\), the Wald distribution becomes more like a standard normal (Gaussian) distribution
limiting
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gamma distribution
log-gamma distribution
If a random variable \(X\) is gamma-distributed with scale \(\alpha\) and shape \(\beta\), then \(Y = log X\) is log gamma-distributed.
transformation
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generalized gamma distribution
gamma distribution
The gamma distribution is a special case of the generalized gamma distribution where \(\gamma=1\)
special case
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Wald distribution
chi-square distribution
If a random variable \(X\) is inverse Gaussian-distributed with mean \(\mu\) and shape parameter \(\lambda\), the \(Y = \lambda(X-\mu)^2/(\mu^2 X)\) has a chi-square distribution
transformation
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exponential distribution
chi-Square distribution
If \(X \sim Exponential(\lambda=1/2)\), then \(X \sim \chi_{2}^{2}\) has a chi-square distribution with 2 degrees of freedom
transformation
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chi-square distribution
Erlang distribution
If \(X \sim \chi^{2}(k)\) with even \(k\), then \(X\) is Erlang distributed with shape parameter \(k/2\) and scale parameter \(1/2\)
special case
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Cauchy distribution
arctangent distribution
The derivative of the arctangent function gives the formula of the Cauchy distribution. Therefore, the arctangent is called the Cauchy cumulative distribution
special case
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exponential distribution
hypoexponential distribution
The hypoexponential distribution is the distribution of a general sum (\(\sum X_i)\) of exponential random variables. Its coefficient of variation is less than one, compared to the exponential distribution, whose coefficient of variation is one
transformation
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Makeham distribution
Gompertz distribution
The Gompertz distribution is a special case of the Makeham distribution where \(\gamma=0\)
special case
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exponential distribution
F distribution
If \(X_1, X_2\) are two independent random variables with exponential distribution with \(\alpha=1\), then \(Y=X_1/X_2\) is an F distribution
special case, transformation
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exponential distribution
hyperexponential distribution
The hyperexponential distribution is the distribution whose density is a weighted sum of exponential densities. Its coefficient of variation is greater than one, compared to the exponential distribution, whose coefficient of variation is one
special case
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IDB distribution
exponential distribution
The exponential distribution is a special case of the IDB distribution where \(\delta=\kappa \to 0\) in the IDB function and \(\alpha=1/ \gamma\) in the exponential function
special case, limiting
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Muth distribution
exponential distribution
The exponential distribution is a particular case of the Muth distribution where \(\alpha=1\) in the exponential function and \(\kappa \to 0\) in the Muth function
special case, limiting
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continuous uniform distribution
exponential power distribution
If \(X\) has a standard uniform distribution, then \(Y=[log(1-log(1-X))/\gamma]^{1/\kappa}\) has an exponential power distribution with parameters \(\lambda\) and \(\kappa\)
transformation
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Laplace distribution
error distribution
The error distribution is a special case of Laplace distribution where \(\alpha_1=\alpha_2\)
special case
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continuous uniform distribution
triangular distribution
If \(X_1, X_2\) are two independent random variables with standard uniform distribution, then \(X = X_1-X_2\) is a standard triangular distribution
transformation
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continuous uniform distribution
power distribution
If X is and independent random variable with standard uniform distribution, then \(X^{1/\beta}\) is a standard power distribution
transformation
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continuous uniform distribution
power distribution
If \(X\) is a standard uniform distribution, then \(X_(n)\) is a standard power distribution
special case
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IDB distribution
Rayleigh distribution
The Rayleigh distribution is a special case of the IDB distribution where \(\delta=2/\alpha, \gamma=0\)
special case, transformation
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Weibull distribution
Rayleigh distribution
The Rayleigh distribution is a special case of the Weibull distribution where \(\beta=1\)
special case
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triangular distribution
triangular distribution
The standard triangular distribution is a special case of the triangular distribution where \(a=-1, b=1, m=0\).
special case
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log-logistic distribution
Lomax distribution
The Lomax distribution is a special case of the Log-logistic distribution where \(\kappa = 1\)
special case
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log-logistic distribution
logistic distribution
If X has a log-logistic distribution with scale parameter \(\alpha\) and shape parameter \(\beta\) then \(Y = log(X)\) has a logistic distribution with location parameter \(log(\alpha)\) and scale parameter \(1 / \beta\).
transformation
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Erlang distribution
exponential distribution
Exponential distribution is a special case of the Erlang distribution where \(n=1\)
special case
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exponential distribution
Erlang distribution
If X has the exponential distribution with parameter \(\alpha\) then \(\sum^n X_i\) has the Erlang distribution with parameters \(n, \alpha\).
transformation
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non-central student t distribution
student t distribution
students t-distribution is a special case of the noncentral students t-distribution where \(\delta=1\)
special case
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logistic exponential distribution
exponential distribution
exponential distribution is a special case of the logistic exponential distribution distribution where \(\beta=1\)
special case
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continuous uniform distribution
Benford distribution
If X has the standard uniform distribution then \(\lfloor 10^X \rfloor\) has the Benford distribution.
transformation
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gamma distribution
inverted gamma distribution
If X has the gamma distribution then \(\frac{1}{X}\) has the inverted gamma distribution.
transformation
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Cauchy distribution
Cauchy distribution
standard Cauchy distribution is a special case of the Cauchy distribution distribution where \(a=0, \alpha=1\)
special case
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Cauchy distribution
hyperbolic secant distribution
If X has the standard Cauchy distribution then \(\log{\frac{\lvert X \rvert}{\pi}}\) has the hyperbolic secant distribution.
transformation
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power-series distribuion
logarithmic distribution
The logarithmic distribution is a particular case of the power-series distribution where \(A(c) = -\log{(1-c)}\)
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
Pascal distribuion
beta-Pascal distribution
When the \(p\propto\beta\) in the pascal distribution, the pascal distribution becomes a beta-pascal
Bayesian
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
Polya distribuion
binomial distribution
The binomial distribution is a particular case of the Polya distribution where \(\beta=1\)
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
Polya distribuion
negative-binomial distribution
The Polya distribuion is a special case of the negative-binomail distribution
special case
ISBN:9780873898317
geometric distribuion
Pascal distribution
If \(X\) has the geometric distribuion with parameter \(p\), then \(\sum\nolimits_{i=1}^n X_i\) has the pascal distribution with parameters \(n, p\).
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
Pascal distribuion
normal distribuion
When \(\mu=n(1-p)\) and \(n\to\infty\) then pascal distribuion becomes normal distribuion.
limiting
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
beta distribution
normal distribution
When \(\beta=\gamma\to\infty\) then beta distribuion becomes normal distribuion.
limiting
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
generalized gamma distribution
log-normal distribution
When \(\beta\to\infty\) then generalized gamma distribuion becomes log-normal distribuion.
limiting
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
chi-square distribution
chi distribution
If \(X\) has the chi-square distribuion with parameter \(n\), then \( \sqrt{X} \) has the chi distribution with parameters \(n\).
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
chi-square distribution
exponential distribution
The exponential distribution is a special case of chi-square distribution with \( \n=2 \).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
exponential distribution
chi-square distribution
The chi-square distribution is a special case of exponential distribution with \( \alpha=2 \).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
beta distribution
inverted beta distribution
If \(X\) has the beta distribuion with parameters \(\beta\) and \(\gamma\), then \( \frac{X}{1-X} \) has the inverted beta distribution with parameters \(\beta\) and \(\gamma\).
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
hypoexponential distribution
Erlang distribution
The Erlang distribution is a special case of hypoexponential distribution with \(\bar{\alpha}=\alpha\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
doubly noncentral t distribution
noncentral t distribution
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
noncentral f distribution
f distribution
When \(\delta\to0\) then noncentral f distribuion becomes f distribuion.
limiting
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
hyperexponential distribution
exponential distribution
The exponential distribution is a special case of hyperexponential distribution with \(\bar{\alpha}=\alpha\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
exponential distribution
Rayleigh distribution
If \(X\) has the exponential distribution with parameter \(\alpha\), then \( X^2 \) has the Rayleigh distribution with parameter \(\alpha\).
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
continuous uniform distribution
Gompertz distribution
If \(X\) has the standard uniform distribution, then \( \frac{log{1-\frac{(\log{X})(\log{k})}{\delta}}}{\log{k}} \) has the Gompertz distribution.
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
error distribution
Laplace distribution
The Laplace distribution is a special case of error distribution with \(a=0\), \(b=\frac{\alpha}{2}\), and \(c=2\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
Laplace distribution
error distribution
The error distribution is a special case of Laplace distribution with \(\alpha_1=\alpha_2\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
continuous uniform distribution
uniform distribution
If \(X\) has the standard uniform distribution, then \( a+(b-a)X \) has the uniform distribution with parameters \(a\) and \(b\).
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
standard power distribution
continuous uniform distribution
The standard uniform distribution is a special case of standard power distribution with \(\beta=1\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
minimax distribution
power distribution
The standard power distribution is a special case of minimax distribution with \(\gamma=1\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
power distribution
power distribution
The standard power distribution is a special case of power distribution with \(\alpha=1\).
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
generalized Pareto distribution
Pareto distribution
If \(X\) has the generalized Pareto distribution with parameters \(\delta\), \(k\), \(\gamma=0\), then \( X+\delta \) has the Pareto distribution with parameters \(k\) and \(\lambda\).
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
extreme-value distribution
Weibull distribution
If \(X\) has the extreme-value distribution, then \( \log{X} \) has the Weibull distribution with the same parameters.
transformation
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
lomax distribution
log-logistic distribution
The log-logistic distribution is a special case of the lomax distribution where \(\kappa = 1\)
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448
TSP distribution
triangular distribution
The triangular distribution is a special case of the TSP distribution where \(n = 2\)
special case
doi:10.1080/07408170590948512
doi:10.1198/000313008X270448