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