4.4.3 The Gamma Distribution

gamma distribution

A continuous random variable $X$ is said to have a gamma distribution if the pdf of $X$ is

$$f\left(x;\alpha ,\beta \right)=\{\begin{array}{cc}\frac{1}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}{x}^{\alpha -1}{e}^{-x/\beta }& x\ge 0\\ 0& \text{ otherwise }\end{array}\text{\hspace{1em}(4.8)}$$

where the parameters $\alpha$ and $\beta$ satisfy $\alpha >0$, $\beta >0$. The standard gamma distribution has $\beta =1$, so the pdf of a standard gamma rv is given by (4.7).

The exponential distribution results from taking $\alpha =1$ and $\beta =1/\lambda$ .

Figure 4.27 (a) Gamma density curves; (b) standard gamma density curves

• Figure 4.27(a) illustrates the graphs of the gamma pdf $f\left(x;\alpha ,\beta \right)$ (4.8) for several $\left(\alpha ,\beta \right)$ pairs
• Figure 4.27(b) presents graphs of the standard gamma pdf.

For the standard pdf,

• when $\alpha \le 1$, $f\left(x;\alpha \right)$ is strictly decreasing as $x$ increases from 0 ;
• when $\alpha >1,f\left(x;\alpha \right)$ rises from 0 at $x=0$ to a maximum and then decreases.

The parameter $\beta$ in (4.8) is a scale parameter, and $\alpha$ is referred to as a shape parameter because changing its value alters the basic shape of the density curve.

The mean and variance of a random variable $X$ having the gamma distribution $f\left(x;\alpha ,\beta \right)$ are

E\left( X\right) = \mu = {\alpha \beta }$$ $$V\left( X\right) = {\sigma }^{2} = \alpha {\beta }^{2}$$ When $X$ is a standard gamma rv, the cdf of $X$,

F\left( {x;\alpha }\right) = {\int }_{0}^{x}\frac{{y}^{\alpha - 1}{e}^{-y}}{\Gamma \left( \alpha \right) }{dy};x > 0 \tag{4.9}

is called the incomplete gamma function - sometimes the incomplete gamma function refers to Expression (4.9) without the denominator $\Gamma \left( \alpha \right)$ in the integrand There are extensive tables of $F\left( {x;\alpha }\right)$ available; in Appendix Table A.4, we present a small tabulation for $\alpha = 1,2,\ldots ,{10}$ and $x = 1,2,\ldots ,{15}$ . [[EX 4.23]] The incomplete gamma function can also be used to compute probabilities involving nonstandard gamma distributions. These probabilities can also be obtained almost instantaneously from various software packages. > [!proposition] > > Let $X$ have a gamma distribution with parameters $\alpha$ and $\beta$ . Then for any $x > 0$ , the cdf of $X$ is given by > $$ > P\left( {X \leq x}\right) = F\left( {x;\alpha ,\beta }\right) = F\left( {\frac{x}{\beta };\alpha }\right) > $$ > where $F\left( {\cdot ;\alpha }\right)$ is the incomplete gamma function. [[EX 4.24]]