An Introduction to Probability and Statistical Inference, 2nd Edition

3.3.2 The Continuous Case

Gamma distribution

For its introduction, a certain function, the so-called Gamma function, is to be defined first. It is shown that the integral ${\displaystyle {\int }_0^{\infty }{y}^{\alpha -1}{e}^{-y}}$ dy is finite for α > 0 and is thus defining a function (in α); namely,

$\begin{array}{cc}\Gamma (\alpha )={\displaystyle {\int }_0^{\infty }{y}^{\alpha -1}{\text{e}}^{-y}\text{d}y,}& \alpha >0.\end{array}$

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This is the Gamma function. By means of the Gamma function, the Gamma distribution is defined as follows through its p.d.f.:

$\begin{array}{cc}f(x)=\frac{1}{\Gamma (\alpha ){\beta }^{\alpha }}{x}^{\alpha -1}{\text{e}}^{-x/\beta },& x>0,\alpha >0,\beta >0;\end{array}$

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α and β are the parameters of the distribution. That the function f integrates to 1 is an immediate ...