Exponential random variable with mean 1/gamma

Question

If $X$ is an exponential random variable with mean $\frac{1}{γ}$, show that $\mathbb{E}[X^k]=\frac{k!}{γ^k},\,\, k=1,2,3,\cdots$

*Use the gamma density function

$\mathbb{E}[X^k]=∫x^{k}γe^{-γx}dx$

I cannot figure out how you incorporate the gamma density function....

Answer 1

Whenever you see factorials popping up mysteriously in integral evaluations, you should always think of the gamma function, and in particular that fact that for any integer $n$

$$n! = \Gamma (n+1) = \int_0^{\infty}x^{n}e^{-x} dx$$

If you make a variable transform $u = \gamma x$ you should be able to massage your integral into something that looks like this.