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