Gamma distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:

1. With a shape parameter k and a scale parameter θ
2. With a shape parameter ${\displaystyle \alpha =k}$ and an inverse scale parameter ${\displaystyle \beta =1/\theta }$ , called a rate parameter.

Gamma

Probability density function
Cumulative distribution function
Parameters	k > 0 shape θ > 0 scale	α > 0 shape β > 0 rate
Support	${\displaystyle x\in (0,\infty )}$	${\displaystyle x\in (0,\infty )}$
PDF	${\displaystyle f(x)={\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-x/\theta }}$	${\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}$
CDF	${\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}$	${\displaystyle F(x)={\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}$
Mean	${\displaystyle k\theta }$	${\displaystyle {\frac {\alpha }{\beta }}}$
Median	No simple closed form	No simple closed form
Mode	${\displaystyle (k-1)\theta {\text{ for }}k\geq 1}$, ${\displaystyle 0{\text{ for }}k<1}$	${\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1{\text{, }}0{\text{ for }}\alpha <1}$
Variance	${\displaystyle k\theta ^{2}}$	${\displaystyle {\frac {\alpha }{\beta ^{2}}}}$
Skewness	${\displaystyle {\frac {2}{\sqrt {k}}}}$	${\displaystyle {\frac {2}{\sqrt {\alpha }}}}$
Ex. kurtosis	${\displaystyle {\frac {6}{k}}}$	${\displaystyle {\frac {6}{\alpha }}}$
Entropy	${\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}$	${\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}$
MGF	${\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}$	${\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }$
CF	${\displaystyle (1-\theta it)^{-k}}$	${\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}$
Fisher information	${\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}$	${\displaystyle I(\alpha ,\beta )={\begin{pmatrix}\psi ^{(1)}(\alpha )&-\beta ^{-1}\\-\beta ^{-1}&\alpha \beta ^{-2}\end{pmatrix}}}$
Method of Moments	${\displaystyle k={\frac {E[X]^{2}}{V[X]}}\quad \quad }$ ${\displaystyle \theta ={\frac {V[X]}{E[X]}}\quad \quad }$	${\displaystyle \alpha ={\frac {E[X]^{2}}{V[X]}}}$ ${\displaystyle \beta ={\frac {E[X]}{V[X]}}}$

In each of these forms, both parameters are positive real numbers.

The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a ${\displaystyle 1/x}$ base measure) for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln X] = ψ(k) + ln θ = ψ(α) − ln β is fixed (ψ is the digamma function).