4.4.2 The Gamma Function

To define the family of gamma distributions, we first need to introduce a function that plays an important role in many branches of mathematics.

gamma function

For $\alpha >0$ , the gamma function $\Gamma \left(\alpha \right)$ is defined by

$$\Gamma \left(\alpha \right)={\int }_0^{\infty }{x}^{\alpha -1}{e}^{-x}dx\text{\hspace{1em}(4.6)}$$

The most important properties of the gamma function are the following:

1. For any $\alpha >1$, $\Gamma \left(\alpha \right)=\left(\alpha -1\right)\cdot \Gamma \left(\alpha -1\right)$
   • via integration by parts
2. For any positive integer, $n$, $\Gamma \left(n\right)=\left(n-1\right)!$
3. $\Gamma \left(1/2\right)=\sqrt{\pi }$

Now let

$$f\left(x;\alpha \right)=\{\begin{array}{cc}\frac{x^{\alpha -1}{e}^{-x}}{\Gamma \left(\alpha \right)}& x\ge 0\\ 0& \text{ otherwise }\end{array}\text{\hspace{1em}(4.7)}$$

Then $f\left(x;\alpha \right)\ge 0$ . Expression (4.6) implies that ${\int }_0^{\infty }f\left(x;\alpha \right)dx=\Gamma \left(\alpha \right)/\Gamma \left(\alpha \right)=1$Thus $f\left(x;\alpha \right)$ satisfies the two basic properties of a pdf.