gamma function

The factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers from 1 to \(n\).

\[ n! = \prod_{k=1}^{n} k \]

The gamma distribution is a continuous probability distribution defined on the positive half-line. Its probability density function is given by

\[
G(x;\alpha,\beta)=
\begin{cases}
\dfrac{1}{\beta^{\alpha}\\,\Gamma(\alpha)}\\, x^{\alpha – 1}\\, e^{-x/\beta} & x>0\\\\[10pt]
0 & \text{elsewhere}
\end{cases}
\]