
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 kth 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

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 $


Quote:
Originally Posted by
craig Ahh thankyou.
Just to make sure you see, I had a big typo up there and I fixed it.

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 :)