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Math Help - Gamma Distribution and Transformations

  1. #1
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    Gamma Distribution and Transformations

    Hi, I have the following question about Gamma Distributions and Transformations. Given the following:



    It is easy to work out that the MGF of X is:



    Now, my doubts lie upon the following. Could someone give me a detailed run through these next two questions?



    As always, any help is greatly appreciated.
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  2. #2
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by aniguchisan View Post
    Hi, I have the following question about Gamma Distributions and Transformations. Given the following:



    It is easy to work out that the MGF of X is:



    Now, my doubts lie upon the following. Could someone give me a detailed run through these next two questions?



    As always, any help is greatly appreciated.
    (a) Multiplicative principle of MGF says:

    M_Z = M_{X_1}\cdot M_{X_2} = (1-\frac{r}{\beta})^{-\alpha_1}(1-\frac{r}{\beta})^{-\alpha_2} = (1-\frac{r}{\beta})^{-(\alpha_1+\alpha_2)}

    Uniqueness implies:

    Z \sim G\Big((\alpha_1+\alpha_2),\beta\Big)
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  3. #3
    MHF Contributor matheagle's Avatar
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    I'm not sure what you want.
    The instructions are quite explicit.
    This is not a difficult problem and it's been solved here several times.
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  4. #4
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by matheagle View Post
    \color{red}{\text{Not sure what you want.}}
    \color{red}{\text{The guide is quite explicit.}}
    \color{red}{\text{This has been solved here.}}
    NOW, it's a Haiku.
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  5. #5
    MHF Contributor matheagle's Avatar
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    if u say so
    I'm not an expert on poetry
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