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Math Help - chi square degrees of freedom

  1. #1
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    chi square degrees of freedom

    If X and Y are independent random variables, X having a chi-square distribution with 3 degrees of freedom, and X+Y having a chi-square dist with 7 degrees of freedom, prove that Y has chi-square dist with 4 degrees of freedom

    Thank you for any help that can be provided
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  2. #2
    MHF Contributor matheagle's Avatar
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    I would use MGFs.

    use X\sim\chi^2_3 and W=X+Y\sim\chi^2_7

    Then M_W(t)=M_X(t)M_Y(t)

    Plug in and by division you have the MGF of Y.
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  3. #3
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    How do you know (aside from practice?) when to use mgf vs. anything else? We just looked at a getting the limiting distribution of

    <br />
\frac {X_r-r*\beta} {\sqrt r*\beta} <br />

    where

    <br />
X\sim\gamma(r,\beta)<br />


    and the solution was best obtained using mgf. Thanks!

    PS: I was able to do the problem, but had the hint do use mgf, so just trying to get a better sense of how to tackle these problems without hints.
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  4. #4
    MHF Contributor matheagle's Avatar
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    That's just the central limit theorem

     \Gamma(\alpha,\beta) is equal in distribution to

    \sum_{i=1}^{\alpha} \Gamma_i(1,\beta)
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