# Gamma Function Question

• Jan 12th 2011, 04:55 AM
craig
Gamma Function Question
I've got the following question in a stats paper, just a bit stuck on how they get their answer:

Suppose that the random variable $\displaystyle X$ follows the exponential distribution $\displaystyle exp(\lambda)$, with p.d.f.

$\displaystyle f(x) = \lambda e^{-\lambda x}$ for $\displaystyle x > 0, \lambda > 0$, 0 otherwise.

Find the find the k-th moment of X, i.e. $\displaystyle E(X^k)$, for some positive integer $\displaystyle k$.

I know that this is $\displaystyle \int^{\infty}_0 \lambda x^k e^{-\lambda x}$, and also that the Gamma Distribution of $\displaystyle \Gamme(k + 1)$ is defined as $\displaystyle \int^{\infty}_0 x^k e^{-x}$, just not sure how to relate the two.

Thanks in advance for any help
• Jan 12th 2011, 05:53 AM
chiph588@
Let $\displaystyle \displaystyle x=\frac t\lambda \implies dx = \frac{dt}\lambda$ and we get $\displaystyle \displaystyle \lambda \int^{\infty}_0 x^k e^{-\lambda x} dx = \int^{\infty}_0 \left(\frac{t}{\lambda}\right)^k e^{-t} dt = \ldots$
• Jan 12th 2011, 06:01 AM
craig
Ahh thankyou.
• Jan 12th 2011, 06:06 AM
chiph588@
Quote:

Originally Posted by craig
Ahh thankyou.

Just to make sure you see, I had a big typo up there and I fixed it.
• Jan 12th 2011, 06:14 AM
craig
Haha actually that does make more sense. I though it was something I was missing at first, was going to have a play around with it, see if I could get where u got to.

Cheers again :)