Statistics/Distributions/Gamma

The Gamma distribution is very important for technical reasons, since it is the parent of the exponential distribution and can explain many other distributions.

The probability distribution function is:

$f_x(x)=\{\begin{array}{cc}\frac{1}{a^p\mathrm{\Gamma }(p)}{x}^{p-1}{e}^{-x/a},& \text{if }x\ge 0\\ 0,& \text{if }x<0\end{array}$

The cumulative distribution function cannot be found unless p=1, in which case the Gamma distribution becomes the exponential distribution. The Gamma distribution of the stochastic variable X is denoted as $X\in \mathrm{\Gamma }(p,a)$.

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter $\alpha =k$ and an inverse scale parameter $\beta =1/\theta$, called a rate parameter:

$g(x;\alpha ,\beta )=K{x}^{\alpha -1}{e}^{-\beta x}\mathrm{f}\mathrm{o}\mathrm{r}x>0.$

where the $K$ constant can be calculated setting the integral of the density function as 1:

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}g(x;\alpha ,\beta )\mathrm{d}t=\underset{0}{\overset{+\mathrm{\infty }}{\int }}K{x}^{\alpha -1}{e}^{-\beta x}\mathrm{d}x=1$

following:

$K\underset{0}{\overset{+\mathrm{\infty }}{\int }}{x}^{\alpha -1}{e}^{-\beta x}\mathrm{d}x=1$

$K=\frac{1}{\underset{0}{\overset{+\mathrm{\infty }}{\int }}{x}^{\alpha -1}{e}^{-\beta x}\mathrm{d}x}$

and, with change of variable $y=\beta x$ :

$\begin{array}{cc}K& =\frac{1}{\underset{0}{\overset{+\mathrm{\infty }}{\int }}\frac{y^{\alpha -1}}{{\beta }^{\alpha -1}}{e}^{-y}\frac{\mathrm{d}y}{\beta }}\\ & =\frac{1}{\frac{1}{{\beta }^{\alpha }}\underset{0}{\overset{+\mathrm{\infty }}{\int }}{y}^{\alpha -1}{e}^{-y}\mathrm{d}y}\\ & =\frac{{\beta }^{\alpha }}{\underset{0}{\overset{+\mathrm{\infty }}{\int }}{y}^{\alpha -1}{e}^{-y}\mathrm{d}y}\\ & =\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}\end{array}$

following:

$g(x;\alpha ,\beta )=x^{\alpha -1}\frac{{\beta }^{\alpha }{e}^{-\beta x}}{\mathrm{\Gamma }(\alpha )}\mathrm{f}\mathrm{o}\mathrm{r}x>0.$