Gamma Function Question

**craig** · January 12th 2011, 04:55 AM

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 $X$ follows the exponential distribution $exp(\lambda)$ , with p.d.f.

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

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

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

Thanks in advance for any help

**chiph588@** · January 12th 2011, 05:53 AM

Let $\displaystyle x=\frac t\lambda \implies dx = \frac{dt}\lambda$ and we get $\displaystyle \lambda \int^{\infty}_0 x^k e^{-\lambda x} dx = \int^{\infty}_0 \left(\frac{t}{\lambda}\right)^k e^{-t} dt = \ldots$

**craig** · January 12th 2011, 06:01 AM

Ahh thankyou.

**chiph588@** · January 12th 2011, 06:06 AM

Originally Posted by craig

Ahh thankyou.

Just to make sure you see, I had a big typo up there and I fixed it.

**craig** · January 12th 2011, 06:14 AM

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

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