Continuous distributions

Click on the function name to produce a plot of the probability density function.

Table 1: Continuous distribution functions

Name	notation	density function	distribution function	mean	variance	mode	median	characteristic fn.	related distributions
Uniform [4,6] $\rule[-3.5mm]{0mm}{10mm}$	$\! \! \begin{array}{l} U(a,b) \ U(0,1) \ U(-1,1) \ \end{array}$	$\! \! \begin{array}{l} p(x)=\frac{1}{a-b}, \rule[-2mm]{3mm}{0mm} x \in [a,b] \\... ...in [0,1] \ p(x)=\frac{1}{2},\rule[-2mm]{3mm}{0mm} x \in [-1,1] \ \end{array}$	$\! \! \begin{array}{l} F(x) = \frac{x - a}{a-b}, \rule[-2mm]{3mm}{0mm} x \in [... ...0,1] \ F(x) = \frac{x+1}{2},\rule[-2mm]{3mm}{0mm} x \in [-1,1] \ \end{array}$	$\! \! \begin{array}{l} \frac{a+b}{2} \ \frac{1}{2} \ 0 \ \end{array}$	$\! \! \begin{array}{l} \frac{b-a}{12} \ \frac{1}{12} \ \frac{1}{6} \ \end{array}$	no mode	$\! \! \begin{array}{l} \frac{a+b}{2} \ \frac{1}{2} \ 0 \ \end{array}$	$\! \! \begin{array}{l} \frac{\exp(i b t) - \exp(i a t) }{i t (b - a)} \ \frac{e^{i t} - 1}{i t} \ \frac{\sin(t)}{t} \ \end{array}$	$-$
Normal/Gaussian [4,6,13] $\rule[-3.5mm]{0mm}{10mm}$	$N(\mu, \sigma^2)$	$p(x) = \frac{1}{ \sigma \sqrt{2 \pi}} \exp{ \left( -\frac{(x-\mu)^2}{2 \sigma^2} \right)}$	$F(x) = \frac{1}{2} + \frac{\mbox{sign}(x-\mu)}{2} \Phi\left(\frac{\vert x-\mu\vert}{\sqrt{2 \sigma^2}}\right)$	$\mu$	$\sigma^2$	$\mu$	$\mu$	$\exp \left( i t \mu - \frac{1}{2} \sigma^2 t^2 \right)$	$-$
Gamma [4,6,11] WWW $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Gamma}(\alpha, \beta)$	$p(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \rule[-2mm]{3mm}{0mm} x>0$	$-$	$\frac{\alpha}{\beta}$	$\frac{\alpha}{\beta^2}$	$\frac{\alpha-1}{\beta}, \; \alpha>1$	$-$	$\left(1 - i \frac{t}{\beta}\right)^{-\alpha}$	$-$
Pearson type III [1,11]	$\mbox{Pearson III}(\alpha, \beta, \mu)$	$p(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} y^{\alpha-1} e^{-\beta y}, y=x-\mu, y>0$	$-$	$\frac{\alpha}{\beta} + \mu$	$\frac{\alpha}{\beta^2}$	$\frac{\alpha - 1}{\beta} + \mu, \; \alpha>1$	$-$	$e^{-i \mu t} \left(1 - i \frac{t}{\beta}\right)^{-\alpha}$	shifted Gamma
Generalized Gamma [4]	$\mbox{Gamma}(\alpha, \beta, \mu, k)$	$p(x) = \frac{\beta^{\alpha \gamma}}{\Gamma(\alpha)} y^{\alpha k-1} e^{-(\beta y)^k}, y=x-\mu, y>0$	$-$	$\frac{\Gamma(\alpha+1/k)}{\beta \Gamma(\alpha)} + \mu, \; \alpha > -\frac{1}{k}$	$\frac{1}{\beta^2} \left\{ \frac{\Gamma(\alpha+2/k)}{\Gamma(\alpha)} - \left[ \frac{\Gamma(\alpha+1/k)}{\Gamma(\alpha)} \right]^2 \right\}, \; \alpha>-2/k$	$\mu + \frac{(\alpha - 1/k)^{1/k}}{\beta}, \; \alpha>1/k$	$-$	$-$	$\! \! \begin{array}{l} \mbox{Gamma}(\alpha, \beta, 0, 1) \sim \mbox{Gamma} \ \... ...mbox{Weibull} \ \mbox{Gamma}(\alpha, \beta, \mu, k) \rightarrow \ \end{array}$
Hyper-exponential $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{HE}(\alpha, \beta, c)$	$p(x) = \frac{c \beta^{\alpha/c}}{\Gamma\left(\frac{\alpha}{c}\right)} x^{\alpha-1} e^{-\beta x^c}, \rule[-2mm]{3mm}{0mm} c,\alpha,\beta,x>0$	$-$	$\frac{\Gamma\left( \frac{1+\alpha}{c} \right)}{\Gamma\left(\frac{\alpha}{c} \right)} \frac{1}{\beta^{\frac{1}{c}}}$	$\frac{\Gamma\left( \frac{2+\alpha}{c} \right)}{\Gamma\left(\frac{\alpha}{c} \right)} \frac{1}{\beta^{\frac{2}{c}}} - E[X]^2$	$-$	$-$	$-$	$\! \! \begin{array}{l} \mbox{HE}(\alpha, \beta, 1) \sim \mbox{Gamma}(\alpha, \b... ...eta, c \right)^q \sim\mbox{HE}\left(\alpha/q, \beta, c/q \right) \ \end{array}$
Exponential [4,6] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Exp}(\lambda)$	$p(x) = \lambda e^{-\lambda x}, \rule[-2mm]{3mm}{0mm} x>0$	$F(x) = 1 - e^{-\lambda x}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$	$0$	$-$	$\left(1 - i \frac{t}{\lambda}\right)^{-1}$	$\mbox{Gamma}(1, \lambda)$
Erlang [4] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Erlang}_n(\lambda)$	$p(x) = \frac{\lambda^{n}}{(n-1)!} x^{n-1} e^{-\lambda x}, \rule[-2mm]{3mm}{0mm} x>0$	$F(x) = 1 - e^{-\lambda x} \sum_{i=0}^{n-1} \frac{(\lambda x)^i}{i!}$	$\frac{n}{\lambda}$	$\frac{n}{\lambda^2}$	$\frac{n-1}{\lambda}$	$-$	$\left(1 - i \frac{t}{\lambda}\right)^{-n}$	$\mbox{Gamma}(n, \lambda)$, $X \sim \sum_{i=1}^{n} \mbox{Exp}(\lambda)$
Chi-square [4,6] $\rule[-3.5mm]{0mm}{10mm}$	$\chi_{\nu}^2$	$p(x) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} x^{\nu/2-1} e^{-x/2}, \rule[-2mm]{3mm}{0mm} x > 0$	$-$	$\nu$	$2 \nu$	$\nu -2$, $\nu \geq 2$	$-$	$(1 - 2 i t)^{-\frac{\nu}{2}}$	$\mbox{Gamma}\left(\frac{\nu}{2},\frac{1}{2} \right)$, $X \sim \sum_{i=1}^{\nu} N_i^2(0,1)$
$\! \! \begin{array}{l} \mbox{Noncentral chi-square} \ \rule[-2mm]{3mm}{0mm} \mbox{(Rayleigh-Rice)} \ \end{array}$ [4] $\rule[-3.5mm]{0mm}{10mm}$	$\chi_{\nu,\delta}^2$	$\! \! \begin{array}{l} p(x) = 2^{-\nu/2} \exp(-(x+\delta)/2) \ \rule[-2mm]{3mm... ...ta^j}{\Gamma(\nu/2 + j) 2^{2j} j!}, \rule[-2mm]{3mm}{0mm} x > 0 \ \end{array}$	$-$	$\nu + \delta$	$2 ( \nu + 2 \delta)$	$-$	$-$	$( 1 - 2 it)^{-\nu/2} \exp\left(\frac{\delta i t}{1 - 2it} \right)$	$-$
Log-normal [4] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Log-}N(\mu, \sigma^2)$	$p(x) = \frac{1}{ \sigma \sqrt{2 \pi}} x^{-1} \exp{ \left( -\frac{(\ln(x)-\mu)^2}{2 \sigma^2} \right)}, \rule[-2mm]{3mm}{0mm} x>0$	$-$	$\exp(\mu + \sigma^2/2)$	$e^{2 \mu + \sigma^2} ( e^{\sigma^2} - 1)$	$e^{\mu - \sigma^2}$	$-$	$-$	$X \sim \exp(N(\mu,\sigma^2))$
Rayleigh [4,14] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Rayleigh}(\sigma^2)$	$p(x) = \frac{1}{\sigma^2} x e^{-x^2/2 \sigma^2}, \rule[-2mm]{3mm}{0mm} x > 0$	$-$	$\sigma \sqrt{\frac{\pi}{2}}$	$\sigma^2 \left(2 - \frac{\pi}{2} \right)$	$\sigma$	$-$	$-$	$\mbox{Weibull}(\sigma,2)$, $X \sim \sqrt{N_1^2(0,\sigma^2)+N_2^2(0,\sigma^2)}$
Rayleigh (generalised) [4] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Rayleigh}_n(\sigma^2)$	$p(x) = \frac{2 x^{n-1}}{(2 \sigma^2)^{n/2} \Gamma(n/2)} e^{-x^2/2 \sigma^2}, \rule[-2mm]{3mm}{0mm} x > 0$	$-$	$\sigma \sqrt{2} \frac{\Gamma((1+n)/2)}{\Gamma(n/2)}$	$-$	$-$	$-$	$-$	$X \sim \sqrt{\sum_{j=0}^{n} N^2_j(0,\sigma^2)}$
Beta [4,6] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Beta}(\alpha, \beta)$	$p(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}, \rule[-2mm]{3mm}{0mm} x \in [0,1]$	$F(x) = \frac{B_x(\alpha, \beta)}{B(\alpha, \beta)}$	$\frac{\alpha}{\alpha + \beta}$	$\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta +1)}$	$\frac{\alpha-1}{\alpha + \beta - 2}$	$-$	$_1F_1(\alpha, \alpha+\beta; i \theta)$	$\mbox{Beta}(\alpha, \beta) = \mbox{Dirichlet}(\alpha, \beta)$
Student-t [6] $\rule[-3.5mm]{0mm}{10mm}$	$t_{\nu}(\mu,\sigma^2)$	$p(x) = \frac{\Gamma((\nu+1)/2)} {\Gamma(\nu/2) \sqrt{\nu \pi \sigma^2}} \left... ...\frac{1}{\nu} \left( \frac{x - \mu}{\sigma} \right)^2 \right)^{-(\nu+1)/2}$	$-$	$\mu, \rule[-2mm]{3mm}{0mm} \nu > 1$	$\frac{\nu}{\nu-2} \sigma^2, \rule[-2mm]{3mm}{0mm} \nu > 2$	$\mu$	$-$	$-$	$\lim_{\nu \rightarrow \infty} t_{\nu}(\mu,\sigma^2) = N(\mu,\sigma^2)$, $\frac{N(0,1)}{\sqrt{\chi^2_{n}/n}}$
Logistic [5, pp.52-53] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Logistic}(\alpha, \beta)$	$-$	$F(x) = \frac{1}{1 + e^{-\alpha t - \beta}},$	$-$	$-$	$-$	$-$	$-$	$-$
Inverse beta [5, p.50] [4] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Inv-Beta}(\alpha, \beta)$	$p(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \frac{x^{\alpha-1}}{ (1+x)^{\alpha+\beta}}, \rule[-2mm]{3mm}{0mm} x > 0$	$-$	$-$	$-$	$-$	$-$	$-$	$\frac{\mbox{Beta}(\alpha, \beta)}{1 -\mbox{Beta}(\alpha,\beta)}$, $\frac{\mbox{Gamma}(1, \alpha)}{\mbox{Gamma}(1, \beta)}$
Inverse gamma [4,6] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Inv-gamma}(\alpha,\beta)$	$p(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{-(\alpha+1)} e^{-\beta/x}, \rule[-2mm]{3mm}{0mm} x>0$	$-$	$\frac{\beta}{\alpha-1}, \rule[-2mm]{3mm}{0mm} \alpha>1$	$\frac{\beta^2}{(\alpha-1)^2 (\alpha-2)}, \rule[-2mm]{3mm}{0mm} \alpha>2$	$\frac{\beta}{\alpha+1}$	$-$	$-$	$1/\mbox{Gamma}(\alpha,\beta)$
Inverse chi-square [6] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Inv-}\chi^2(\nu)$	$p(x) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} x^{-(\nu/2+1)} e^{-1/2x}, \rule[-2mm]{3mm}{0mm} x>0$	$-$	$\frac{1}{\nu-2}, \rule[-2mm]{3mm}{0mm} \nu>2$	$\frac{2}{(\nu-2)^2 (\nu-4)}, \rule[-2mm]{3mm}{0mm} \nu>4$	$\frac{1}{\nu+2}$	$-$	$-$	$\mbox{Inv-gamma}(\nu/2, 1/2)$
$\! \! \begin{array}{l} \mbox{Scaled inverse} \ \rule[-2mm]{3mm}{0mm} \rule[-2mm]{3mm}{0mm} \mbox{chi-square} \ \end{array}$ [6] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Inv-}\chi^2(\nu, s^2)$	$p(x) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} s^{\nu} x^{-(\nu/2+1)} e^{-\nu s^2/2x}, \rule[-2mm]{3mm}{0mm} x>0$	$-$	$\frac{\nu}{\nu-2} s^2$	$\frac{2 \nu^2}{(\nu-2)^2 (\nu-4)} s^4$	$\frac{\nu}{\nu+2} s^2$	$-$	$-$	$\mbox{Inv-gamma}(\nu/2, s^2 \nu/2)$
Cauchy [4,14,15,16] $\rule[-3.5mm]{0mm}{10mm}$	$\! \! \begin{array}{l} \mbox{Cauchy}(\mu,b) \ \mbox{Cauchy}(0,1) \ \end{array}$	$\! \! \begin{array}{l} p(x) = \frac{1}{\pi b \left\{ 1+ \left( \frac{x-\mu}{b} \right)^2 \right\} } \ p(x) = \frac{1}{\pi (1+x^2)} \ \end{array}$	$-$	$\! \! \begin{array}{l} \mbox{DNE} \ \mbox{DNE} \ \end{array}$	$\! \! \begin{array}{l} \mbox{DNE} \ \mbox{DNE} \ \end{array}$	$\! \! \begin{array}{l} \mu \ 0 \ \end{array}$	$\! \! \begin{array}{l} \mu \ 0 \ \end{array}$	$\! \! \begin{array}{l} e^{i t \mu} e^{-\vert t\vert b} \ e^{-\vert t\vert} \ \end{array}$	$\! \! \begin{array}{l} t_1(\mu, \sigma^2) \ \ \end{array}$
Laplace [4] $\rule[-3.5mm]{0mm}{10mm}$ [16]	$\mbox{Laplace}(\mu,b)$	$p(x) = \frac{1}{2b} \exp\left(\frac{-\vert x-\mu\vert}{b}\right)$	$-$	$\mu$	$2 b^2$	$\mu$	$\mu$	$\frac{\exp(i \mu t)}{(1+b^2 t^2)}$	$-$
Double exponential $\rule[-1.5mm]{0mm}{6mm}$ [16]	$\mbox{Laplace}(0,1)$	$p(x) = \frac{1}{2} e^{-\vert x\vert}$	$-$	$0$	$2$	$0$	$0$	$\frac{1}{1+t^2}$	$-$
Triangular (Simpson) [4,14,16] $\rule[-3.5mm]{0mm}{10mm}$	$\! \! \begin{array}{l} \mbox{Triangular}(a,b,c) \ \ \mbox{Triangular}(-1,1,0) \ \end{array}$	$\! \! \begin{array}{l} p(x) = \left\{ \begin{array}{l} \frac{2(x-a)}{(b-a)(c-... ... \ p(x) = 1 - \vert x\vert, \rule[-2mm]{3mm}{0mm} x \in [-1,1] \ \end{array}$	$-$	$\! \! \begin{array}{l} \frac{a+b+c}{3} \ \ 0 \ \end{array}$	$\! \! \begin{array}{l} \frac{a^2 + b^2 + c^2 - ab - ac -bc}{18} \ \ 1/6 \ \end{array}$	$\! \! \begin{array}{l} c \ \ 0 \ \end{array}$	$-$	$\! \! \begin{array}{l} - \ \ 2 \left( \frac{1-\cos(t)}{t^2} \right) \ \end{array}$	$\! \! \begin{array}{l} \ \ U(-1/2,1/2) + U(-1/2,1/2) \ \end{array}$
Anon [16] $\rule[-3.5mm]{0mm}{10mm}$		$p(x) = \frac{1 - \cos(x)}{\pi x^2}$	$-$	$-$	$-$	$-$	$-$	$(1 - \vert t \vert) I_{[-1,1]}(t)$	$-$
Maxwell [14] $\rule[-3.5mm]{0mm}{10mm}$		$p(x) = \sqrt{ \frac{2}{\pi} } \frac{ (x - x_0)^2 }{a^3} \exp\left( - \frac{ (x-x_0)^2 }{2 a^2} \right), \rule[-2mm]{3mm}{0mm} x \geq x_0$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
$\! \! \begin{array}{l} \mbox{Extreme value I \cite{leadbetter83:_extremes,Eva... ...3mm}{0mm} \mbox{doubly exponential \cite{Abramowitz_and_Stegun}} \ \end{array}$	$\mbox{EV}_{I}(a,b)$	$p(x) = \frac{1}{b} \exp\left( \frac{-(x-a)}{b} \right) \exp\left\{ -\exp\left( \frac{-(x-a)}{b} \right) \right\}$	$F(x) = \exp\left\{ -\exp\left( \frac{-(x-a)}{b} \right) \right\}$	$a - b \Gamma'(1)$	$b \pi^2/6$	$a$	$a - b \log \log 2$	$\exp(iat) \Gamma(1 - ibt)$	$EV(a,b) = a - \log(\mbox{Exp}(1/b))$
Gumbel $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{EV}_{I}(0,1)$	$p(x) = \exp\left( -x \right) \exp\left\{ -\exp\left( -x \right) \right\}$	$F(x) = \exp\left\{ -\exp\left( -x \right) \right\}$	$\Gamma'(1)$	$\pi^2/6$	$0$	$\log \log 2$	$\Gamma(1 - i t)$	particular case of extreme value
Extreme value II [12]	$\mbox{EV}_{II}(\alpha)$	$-$	$F(x) = \exp\left\{ - x^{-\alpha}\right\}, x \geq 0, \alpha >0$	$-$	$-$	$-$	$-$	$-$	maximum of type II series, e.g. Pareto
Extreme value III [12]	$\mbox{EV}_{III}(\alpha)$	$-$	$F(x) = \exp\left\{ - (-x)^{\alpha}\right\}, x \leq 0, \alpha >0$	$-$	$-$	$-$	$-$	$-$	maximum of type III series
Weibull [4,14] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Weibull}(b,c)$	$p(x) = \frac{ c x^{c-1} }{b^c} \exp\left[ - \left(\frac{x}{b}\right)^c \right], \rule[-2mm]{3mm}{0mm} x \geq 0$	$F(x) = 1 - \exp\left[ - \left(\frac{x}{b}\right)^c \right]$	$b \Gamma\left( \frac{c+1}{c} \right)$	$b^2 \left[ \Gamma\left( \frac{c+2}{c} \right) - \Gamma\left( \frac{c+1}{c} \right)^2 \right]$	$\begin{array}{ll} b \left(1 - \frac{1}{c}\right)^{1/c}, & c \geq 1 \\ 0, & c \leq 1 \\ \end{array}$	$b (\log 2)^{1/2}$	$-$	$-$
Pareto [4,15] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Pareto}(a,c)$	$p(x) = \frac{ c a^{c} }{x^{c+1}}, \rule[-2mm]{3mm}{0mm} x \geq a$	$F(x) = 1 - \left( \frac{a}{x} \right)^c$	$\frac{ca}{c-1}, \rule[-2mm]{3mm}{0mm} c>1$	$\frac{c a^2}{ (c-1)^2 (c-2) }, \rule[-2mm]{3mm}{0mm} c > 2$	$a$	$2^{1/c} a$	$-$	$X \sim exp(a \mbox{Exp}(c))$
Pareto II $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{ParetoII}(a,c)$	$p(x) = \frac{ c a^{c} }{(x+a)^{c+1}}, \rule[-2mm]{3mm}{0mm} x \geq 0$	$F(x) = 1 - \left( \frac{a}{x+a} \right)^c$	$\frac{a}{c-1}, \rule[-2mm]{3mm}{0mm} c>1$	$-$	$0$	$-$	$-$	shifted version of Pareto
Pareto III $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{ParetoIII}(a,c)$	$-$	$F(x) = 1 - \frac{a^c}{a^c + x^c}$	$\frac{a \pi c}{\sin \pi c}, \rule[-2mm]{3mm}{0mm} c>1$	$-$	$-$	$a$	$-$	shifted version of Pareto
Bessel $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{Bessel}(\alpha)$	$p(x) = \frac{ \alpha e^{-\alpha} I_{\alpha}(x) }{x}, \rule[-2mm]{3mm}{0mm} x > 0$	$-$	$-$	$-$	$-$	$-$	$\left[ 1 + it - \sqrt{ (1+it)^2 - 1} \right]^{\alpha}$	$-$
Arcsine [10]		$\mbox{$\rule[-3.5mm]{0mm}{10mm}$}p(x) = \frac{1}{ \pi \sqrt{x(1-x)} }, \rule[-2mm]{3mm}{0mm} x \in (0,1)$	$F(x) = \frac{2}{\pi} \arcsin(\sqrt{x})$	$-$	$-$	$-$	$-$	$-$	$\mbox{Beta} \left(\frac{1}{2}, \frac{1}{2} \right)$
Generalized Arcsine [5, pp.470-471]		$\mbox{$\rule[-3.5mm]{0mm}{10mm}$}p(x) = \frac{\sin(\pi \alpha)}{\pi} x^{-\al... ...1}, \rule[-2mm]{3mm}{0mm} x \in (0,1) \rule[-2mm]{3mm}{0mm} \alpha \in (0,1)$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
Hyperbolic cosine [5]		$\mbox{$\rule[-3.5mm]{0mm}{10mm}$}p(x) = \frac{ 1}{ \pi \cosh(x)}, \; \; \cosh(x) = \frac{1}{2} (e^x + e^{-x})$	$F(x) = 1 - \frac{2}{\pi} \arctan(e^{-x})$	$0$	$-$	$-$	$-$	$\frac{1}{\cosh(\pi t/2)}$	$-$
Fisher-Snedecor (Variance ratio) [4] $\rule[-3.5mm]{0mm}{10mm}$	$\mbox{F}_{\nu, \omega}$	$p(x) = \frac{\Gamma[(\nu+\omega)/2] (\nu/\omega)^{\nu/2} x^{(\nu-2)/2}} {\G... ... + (\nu/\omega) x \right)^{(\nu+\omega)/2} }, \rule[-2mm]{3mm}{0mm} x \geq 0$	$-$	$\frac{\omega}{\omega-2}, \rule[-2mm]{3mm}{0mm} \omega > 2$	$\frac{2 \omega^2 ( \nu + \omega - 2)} { \nu (\omega - 2)^2 (\omega - 4) } , \rule[-2mm]{3mm}{0mm} \omega > 4$	$\frac{\omega (\nu - 2)}{\nu ( \omega + 2) }, \rule[-2mm]{3mm}{0mm} \nu > 2$	$-$	$-$	$\frac{\chi^2_{\nu}/\nu}{\chi^2_{\omega}/\omega}$
$\! \! \begin{array}{l} \mbox{Inverse Gaussian}\cite{Evans_et_al(1993)} \ \rule[-2mm]{3mm}{0mm} \mbox{Wald} \ \end{array}$ $\rule[-3.5mm]{0mm}{10mm}$	$\! \! \begin{array}{l} \mbox{IN}(\mu, \lambda) \ \mbox{IN}(1, \lambda) \ \end{array}$	$\! \! \begin{array}{l} p(x) = \sqrt{\frac{\lambda}{2 \pi x^3}} \exp \left( \f... ...pi x^3}} \exp \left( \frac{ - \lambda (x - 1)^2}{2 x} \right) \ \end{array}$	$-$	$\! \! \begin{array}{l} \mu \ 1 \ \end{array}$	$\! \! \begin{array}{l} \mu^3/\lambda \ 1/\lambda \ \end{array}$	$\! \! \begin{array}{l} \mu \left[ \left( 1 + \frac{ 9 \mu^2}{4 \lambda^2} \ri... ...c{9}{4 \lambda^2} \right)^{1/2} - \frac{3}{2 \lambda} \right] \ \end{array}$	$-$	$\exp \left[ \frac{\lambda}{\mu} \left\{ 1 - \left( 1 - \frac{2 \mu^2 i t}{\lambda} \right)^{1/2} \right\} \right]$	$\lim_{\lambda \rightarrow \infty} X = N(0,1)$
Kovalenko distribution [7] $\rule[-3.5mm]{0mm}{10mm}$	$R_{\nu-1}$,	$-$	$1 - \frac{1}{\pi} \sum_{n=1}^{H} (-1)^{n-1} \frac{\Gamma(n (\nu-1)) \sin(n (\nu-1) \pi) }{t^{n (\nu-1)}} + O(x^{-(H+1)(\nu-1)}), H=1,2,\ldots$	$-$	$-$	$-$	$-$	$-$	$-$
Kovalenko distribution [3] $\rule[-3.5mm]{0mm}{10mm}$	$R_{1/2}$,	$-$	$1 - \frac{2}{\sqrt{\pi}} e^{x} \mbox{Erfn}(x^{1/2})$	$-$	$-$	$-$	$-$	$-$	$-$