Gamma Distribution (Gamma₍α,β₎) ○◂｜_append｜___｜20260628182425-01-•

Gamma	…	…	…
Probability density function	…	…	…
Cumulative distribution function	…	…	…
Parameters	• α > 0 shape • θ > 0 scale	• α > 0 shape • β > 0 rate
Support	$x\in [0,\infty )$	$x\in [0,\infty )$
PDF	$f(x)=\frac{1}{\Gamma (\alpha ){\theta }^{\alpha }}{x}^{\alpha -1}{e}^{-x/\theta }$	$f(x)=\frac{{\beta }^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}{e}^{-\beta x}$
CDF	$F(x)=\frac{1}{\Gamma (\alpha )}\gamma \left(\alpha ,\frac{x}{\theta }\right)$	$F(x)=\frac{1}{\Gamma (\alpha )}\gamma (\alpha ,\beta x)$
Mean	$\alpha \theta$	$\frac{\alpha }{\beta }$
Median	Simple closed form does not exist	Simple closed form does not exist
Mode	$(\alpha -1)\theta \text{ for }\alpha \ge 1$, $0\text{ for }\alpha <1$	$\frac{\alpha -1}{\beta }\text{ for }\alpha \ge 1\text{, }0\text{ for }\alpha <1$
Variance	$\alpha {\theta }^2$	$\frac{\alpha }{{\beta }^2}$
Skewness	$\frac{2}{\sqrt{\alpha }}$	$\frac{2}{\sqrt{\alpha }}$
Excess kurtosis	$\frac{6}{\alpha }$	$\frac{6}{\alpha }$
Entropy	$\begin{array}{rl}\alpha & +\ln \theta +\ln \Gamma (\alpha )\\ & +(1-\alpha )\psi (\alpha )\end{array}$	$\begin{array}{rl}\alpha & -\ln \beta +\ln \Gamma (\alpha )\\ & +(1-\alpha )\psi (\alpha )\end{array}$
MGF	$(1-\theta t{)}^{-\alpha }\text{ for }t<\frac{1}{\theta }$	${\left(1-\frac{t}{\beta }\right)}^{-\alpha }\text{ for }t<\beta$
CF	$(1-\theta it{)}^{-\alpha }$	${\left(1-\frac{it}{\beta }\right)}^{-\alpha }$
Fisher information	$I(\alpha ,\theta )=\left(\begin{array}{cc}{\psi }^{(1)}(\alpha )& {\theta }^{-1}\\ {\theta }^{-1}& \alpha {\theta }^{-2}\end{array}\right)$	$I(\alpha ,\beta )=\left(\begin{array}{cc}{\psi }^{(1)}(\alpha )& -{\beta }^{-1}\\ -{\beta }^{-1}& \alpha {\beta }^{-2}\end{array}\right)$
Method of moments	$\alpha =\frac{E[X{]}^2}{V[X]},$ $\theta =\frac{V[X]}{E[X]}$	$\alpha =\frac{E[X{]}^2}{V[X]},$ $\beta =\frac{E[X]}{V[X]}$

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