Gamma Distribution

Gamma

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \scriptstyle k\;>\;0}$ shape ${\displaystyle \scriptstyle \theta \;>\;0\,}$ scale
Support	${\displaystyle \scriptstyle x\;\in \;(0,\,\infty )\!}$
PDF	${\displaystyle \scriptstyle {\frac {1}{\Gamma (k)\theta ^{k}}}x^{k\,-\,1}e^{-{\frac {x}{\theta }}}\,\!}$
CDF	${\displaystyle \scriptstyle {\frac {1}{\Gamma (k)}}\gamma \left(k,\,{\frac {x}{\theta }}\right)\!}$
Mean	${\displaystyle \scriptstyle \operatorname {E} [X]=k\theta \!}$ ${\displaystyle \scriptstyle \operatorname {E} [\ln X]=\psi (k)+\ln(\theta )\!}$ (see digamma function)
Median	No simple closed form
Mode	${\displaystyle \scriptstyle (k\,-\,1)\theta {\text{ for }}k\;>\;1\,\!}$
Variance	${\displaystyle \scriptstyle \operatorname {Var} [X]=k\theta ^{2}\,\!}$ ${\displaystyle \scriptstyle \operatorname {Var} [\ln X]=\psi _{1}(k)\!}$ (see trigamma function )
Skewness	${\displaystyle \scriptstyle {\frac {2}{\sqrt {k}}}\,\!}$
Ex. kurtosis	${\displaystyle \scriptstyle {\frac {6}{k}}\,\!}$
Entropy	${\displaystyle \scriptstyle {\begin{aligned}\scriptstyle k&\scriptstyle \,+\,\ln \theta \,+\,\ln[\Gamma (k)]\\\scriptstyle &\scriptstyle \,+\,(1\,-\,k)\psi (k)\end{aligned}}}$

The Gamma distribution is very important for technical reasons, since it is the parent of the exponential distribution and can explain many other distributions.

The probability distribution function is:

${\displaystyle f_{x}(x)={\begin{cases}{\frac {1}{a^{p}\Gamma (p)}}x^{p-1}e^{-x/a},&{\mbox{if }}x\geq 0\\0,&{\mbox{if }}x<0\end{cases}}\quad a,p>0}$

Where ${\displaystyle \Gamma (p)=\int _{0}^{\infty }t^{p-1}e^{-t}\,dt\,}$ is the Gamma function. The cumulative distribution function cannot be found unless p=1, in which case the Gamma distribution becomes the exponential distribution. The Gamma distribution of the stochastic variable X is denoted as ${\displaystyle X\in \Gamma (p,a)}$.

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter ${\displaystyle \alpha =k}$ and an inverse scale parameter ${\displaystyle \beta =1/\theta }$, called a rate parameter:

${\displaystyle g(x;\alpha ,\beta )=Kx^{\alpha -1}e^{-\beta \,x}\ \mathrm {for} \ x>0\,\!.}$

where the ${\displaystyle K}$ constant can be calculated setting the integral of the density function as 1:

${\displaystyle \int _{-\infty }^{+\infty }g(x;\alpha ,\beta )\mathrm {d} t\,=\int _{0}^{+\infty }Kx^{\alpha -1}e^{-\beta \,x}\mathrm {d} x\,=1}$

following:

${\displaystyle K\int _{0}^{+\infty }x^{\alpha -1}e^{-\beta \,x}\mathrm {d} x\,=1}$

${\displaystyle K={\frac {1}{\int _{0}^{+\infty }x^{\alpha -1}e^{-\beta \,x}\mathrm {d} x}}}$

and, with change of variable ${\displaystyle y=\beta x}$ :

${\displaystyle {\begin{aligned}K&={\frac {1}{\int _{0}^{+\infty }{\frac {y^{\alpha -1}}{\beta ^{\alpha -1}}}e^{-y}{\frac {\mathrm {d} y}{\beta }}}}\\&={\frac {1}{{\frac {1}{\beta ^{\alpha }}}\int _{0}^{+\infty }y^{\alpha -1}e^{-y}\mathrm {d} y}}\\&={\frac {\beta ^{\alpha }}{\int _{0}^{+\infty }y^{\alpha -1}e^{-y}\mathrm {d} y}}\\&={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\end{aligned}}}$

following:

${\displaystyle g(x;\alpha ,\beta )=x^{\alpha -1}{\frac {\beta ^{-\alpha }\,e^{-\beta \,x}}{\Gamma (\alpha )}}\ \mathrm {for} \ x>0\,\!.}$