# hard proof- gamma distribution

• October 29th 2010, 06:18 AM
holly123
hard proof- gamma distribution
if X is an exponential random variable with mean 1/lambda, show that the expected value of x^k = k!/lambda^k
Hint: make use of the gamma density function
• October 29th 2010, 06:53 AM
TheEmptySet
The density function for an exponential is

$f(x)=\lambda e^{-\lambda^x}$

so

$\mathbb{E}[x^k]=\int_{0}^{\infty}x^{k}f(x)dx =\lambda\int_{0}^{\infty}x^{k}e^{-\lambda x}dx$

let $u=\lambda x$ and the resulting integral is the gamma function.

$\Gamma(k+1=k!)$

Gamma function - Wikipedia, the free encyclopedia