Example 4: Gamma distribution

A positive random variable is gamma distributed $X\sim Gamma(\alpha,\beta)$ when

$$f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}{\rm e}^{-\beta x}$$ (4.8)

where $\alpha,\beta>0$. The parameter $\alpha$ is known as the shape parameter and determines the shape (skewness) of the distribution, whereas parameter $\beta$ is known as the inverse scale parameter and determines the scale/width of the distribution. The population mean $E(X)=\alpha/\beta$ and the population variance $var(X)=\alpha/\beta^2$. The coefficient of variation, $\sigma/\mu=1/\sqrt{\alpha}$, provides a simple (moment) method estimate of the shape parameter. The mode of the distribution is less than the mean and located at $(\alpha-1)/\beta$ when $\alpha>1$. For $\alpha\leq 1$, the Gamma density is inverse J-shaped with the mode at $x=0$.

The gamma distribution is useful for describing positively skewed positive variables such as rainfall totals. A nice additive property of gamma distributed variables is that if $X_1$ and $X_2$ are independent with $X_1\sim Gamma(\alpha_1,\beta)$ and $X_2\sim Gamma(\alpha_1,\beta)$, then $X_1+X_2\sim Gamma(\alpha_1+\alpha_2,\beta)$. For example, the sum $S$ of $n$ independent rainfall totals distributed as $X\sim Gamma(\alpha,\beta)$ will also be Gamma distributed as $S\sim Gamma(n\alpha,\beta)$.

Several commonly used distributions are special cases of the gamma distributions. The exponential distribution $X\sim Expon(\beta)$ is the special case of the Gamma distribution when $\alpha=1$ i.e. $X\sim Gamma(1,\beta)$. The special case $X\sim Gamma(n/2,1/2)$ is also known as the chi-squared distribution $X\sim \chi^2_n$ with $n$ degrees of freedom. The chi-squared distribution describes the distribution of the sum of squares of $n$ independent standard normal variables, and so for example, the sample variance of $n$ independent normal variates is distributed as $s^2/\sigma^2\sim \chi^2_{n-1}$ (there are $n-1$ degrees of freedom rather than $n$ since one is lost in estimating the sample mean).