Distribution

Gamma Distribution

The gamma distribution is a continuous probability distribution that is widely used because it’s very related to the Exponential Distribution and Exponential Distribution and Normal Distribution.

You should first be familiar with the Gamma Function.

Gamma Distribution (Definition)

A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha >0$ and $\lambda >0$, shown as $X\sim Gamma(\alpha ,\lambda )$, if its PDF is given by

$f_X(x)=\{\begin{array}{ll}\frac{{\lambda }^{\alpha }{x}^{\alpha -1}{e}^{-\lambda x}}{\Gamma (\alpha )}& x>0\\ 0& x\le 0\end{array}$

Notice that $Gamma(1,\lambda )\sim Exponential(\lambda )$. More generally, if you sum $n$ independent $Exponential(\lambda )$ random variables, you get $Gamma(n,\lambda )$ random variables.

There are two different parameterizations in common use:

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

Related

The Exponential Distribution, Exponential Distribution, Erlang Distribution, and Exponential Distribution, Erlang Distribution, and Chi-Squared Distribution are special cases of the Gamma Distribution.