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Thread: Gamma Function Question

  1. #1
    Super Member craig's Avatar
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    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
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    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 $
    Last edited by chiph588@; Jan 12th 2011 at 06:06 AM. Reason: Major typo!
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  3. #3
    Super Member craig's Avatar
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    Ahh thankyou.
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by craig View Post
    Ahh thankyou.
    Just to make sure you see, I had a big typo up there and I fixed it.
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  5. #5
    Super Member craig's Avatar
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    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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