Home page for accesible maths 7.2 Exponential Distribution:

{\rm Exp}(\beta)

7.2.1 Lack of memory 7.2.3 Expectation and variance of

{\rm Exp}(\beta)

rvs

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7.2.2 The Gamma function

The integral required to obtain the expected value and variance of a rv with an exponential distribution will occur several times in this chapter. We first define a slightly simplified form, which occurs in many areas of mathematics, and discover several of its properties.

The Gamma function, $\Gamma(\alpha)$ , is defined as $\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}\exp(-t)\,dt$

Firstly we note that

\Gamma(1)={\color[rgb]{1,1,1}{\int_{0}^{\infty}\exp(-t)\,dt=\left[-\exp(-t)% \right]_{0}^{\infty}=1.}}

Proposition 7.8.

For $\alpha>0$ ,

\displaystyle\Gamma(\alpha+1)=\alpha\,\Gamma(\alpha).

Proof.

We use integration by parts:

$\displaystyle\Gamma(\alpha+1)$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}t^{\alpha}\exp(-t)\,dt$
	$\displaystyle=$	$\displaystyle{\color[rgb]{1,1,1}{[t^{\alpha}(-1)\exp(-t)]_{0}^{\infty}+\int_{0% }^{\infty}\alpha t^{\alpha-1}\exp(-t)\,dt}}$
	$\displaystyle=$	$\displaystyle{\color[rgb]{1,1,1}{0-0+\alpha\,\Gamma(\alpha)\mbox{ for }\alpha>% 0.}}$

∎

Since $\Gamma(1)=1$ , we have that $\Gamma(2)=1$ , $\Gamma(3)=2\Gamma(2)=2$ , $\Gamma(4)=3\Gamma(3)=6,\dots$ . By induction we can easily see that for positive integers $n$ , $\Gamma(n+1)=n!$ .