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Math Help - Gamma distribution question with n independent variables

  1. #1
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    Gamma distribution question with n independent variables

    I need a some help with the following statement:

    Let Y_1,Y_2,...,Y_n be independent r.v., where Y_i has a gamma distribution, with parameters \alpha_i and \beta (where \beta is fixed and the values of \alpha are different). Prove that U=Y_1+Y_2+...+Y_n has a gamma distribution with parameters \alpha_1+\alpha_2+...+\alpha_n and \beta

    so far I have:

    U=Y_1+Y_2+...+Y_n

    m_U(t) = m_{Y_1}(t)\times m_{Y_2}(t)\times ... \times m_{Y_n}(t)

    m_U(t) = (1-\beta)^{-\alpha_1} \times (1-\beta)^{-\alpha_2} \times ... \times (1-\beta)^{-\alpha_n}

    m_U(t)=\prod^{n}_{i=1} (1-\beta)^{-\alpha_i}= \

    at which point I don't know how to continue it.
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  2. #2
    MHF Contributor

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    Why not write: <br />
m_U(t)=\prod^{n}_{i=1} (1-\beta)^{-\alpha_i}= (1-\beta)^{-\sum_{i=1}^n\alpha_i}<br />
?

    (By the way, it is not straightforward to show that the moment generating function of a Gamma distribution is (1-\beta)^{-\alpha}, and the most elementary proof of your statement is by change of variable in multiple integrals. But if you were given the moment generating function, you're definitely right using it, since this is way quicker.)
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