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Math Help - how to show their sum is another distribution

  1. #1
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    how to show their sum is another distribution

    Xi idd exponential distribution, show sum of n Xi is gamma distribution.

    i think this is typical question. but i got no idea at all. So you try to show p.d.f.? or other ways?
    Last edited by pengchao1024; October 12th 2009 at 03:03 PM.
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  2. #2
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    Quote Originally Posted by pengchao1024 View Post
    give me a hint guys:

    Xi iid some known distribution, how to show their sum is another distribution
    i just want a hint on what to do to show that.
    More details would help.
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  3. #3
    MHF Contributor matheagle's Avatar
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    Use the product of the MGFs.
    The sum of INDEPENDENT gamma's with the same BETA

    So if X_i\sim\Gamma (\alpha_i, \beta) then

    \sum_{i=1}^nX_i\sim\Gamma (\sum_{i=1}^n\alpha_i, \beta)

    That's also how you prove the sum of independent chi-squares is a \chi^2.

    You just add the dfs.

    BUT you need independence.

    I'm working on a random stopping boot strap problem where the chi-squares are NOT independent.
    Anyone want to help me? lol
    Last edited by matheagle; October 12th 2009 at 10:39 PM.
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  4. #4
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    ok

    how can u find the density function of Gamma distribution by using convolution?
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  5. #5
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by pengchao1024 View Post
    how can u find the density function of Gamma distribution by using convolution?

    painfully, that's an n-fold integral
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  6. #6
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    ok

    well, would you do it like first two variables for example to show the ideas of using induction
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