The Gamma Distribution

A continuous random variable $X$ has the Gamma distribution with shape parameter $\alpha > 0$ and scale parameter $\beta > 0,$ written $X \sim \Gamma(\alpha, \beta),$ if its probability density function is

$$f(x) = \dfrac{1}{\Gamma(\alpha)\,\beta^{\alpha}}\, x^{\alpha - 1}\, e^{-x/\beta}, \qquad x > 0,$$

and $f(x) = 0$ for $x \le 0.$ Here $\Gamma(\alpha)$ is the Gamma function.

The Gamma distribution generalizes the exponential distribution: when $\alpha = 1,$ we have $\Gamma(1) = 1$ and $x^{0} = 1,$ so the PDF reduces to $f(x) = \dfrac{1}{\beta} e^{-x/\beta},$ the exponential distribution with mean $\beta.$

To illustrate, suppose $X \sim \Gamma(\alpha = 3, \beta = 1).$ Recall that $\Gamma(3) = 2! = 2.$ The density at $x = 2$ is

$$\begin{align*} f(2) &= \dfrac{1}{\Gamma(3) \cdot 1^{3}} \cdot 2^{3-1} \cdot e^{-2/1} \\[3pt] &= \dfrac{1}{2} \cdot 4 \cdot e^{-2} \\[3pt] &= 2 e^{-2}. \end{align*}$$