The Gamma Distribution

Posted by Beetle B. on Tue 06 June 2017

The problem with the normal distribution is that it is symmetric. The Gamma distribution is useful for skewed distributions.

Gamma Function

For \(\alpha>0\):

\begin{equation*} \Gamma\left(\alpha\right)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}\ dx \end{equation*}

It has the following properties:

  • \(\Gamma(\alpha)=(\alpha-1)\Gamma(\alpha-1)\) for \(\alpha>1\)
  • \(\Gamma(n)=(n-1)!\)
  • \(\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\)

Probability Distribution Function

For \(\alpha,\beta>0\):

\begin{equation*} f(x;\alpha,\beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\beta^{\alpha}\Gamma(\alpha)} \end{equation*}

for \(x\ge0\)

The standard Gamma distribution has \(\beta=1\)

\(\beta\) is essentially a scale parameter - it squeezes or widens.

The cdf of the standard distribution is called the incomplete gamma function.

\begin{equation*} P(X\le x)=F(x;\alpha,\beta)=F\left(\frac{x}{\beta};\alpha\right) \end{equation*}

Mean, Variance

Mean is \(\alpha\beta\)

Variance is \(\alpha\beta^{2}\)