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Math Help - Gamma Function Proofs - having trouble

  1. #1
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    Gamma Function Proofs - having trouble

    I feel like I should be understanding these, but just can't make any headway:

    1) If X~gamma(alpha, theta), then show that cX~gamma(alpha, theta/c).

    and

    2) If X1, X2, ..., Xn are independent and ~Normal(mu, sigma^2), show that (1/sigma^2)(X1^2 + X2^2 + ... + Xn^2)~gamma(n/2, 1/2).

    Any suggestions or help much appreciated.
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  2. #2
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    Quote Originally Posted by 1of42 View Post
    I feel like I should be understanding these, but just can't make any headway:

    1) If X~gamma(alpha, theta), then show that cX~gamma(alpha, theta/c). Mr F says: This is wrong. {\color{red}cX} ~ {\color{red}\text{gamma}(\alpha, c \theta)}.

    [snip]
    There are several aproaches. The easiest is to calculate the moment generating function of cX.

    Note that if Y = cX and m_X(t) is the moment generating function of X then m_Y(t) = m_X(ct).

    Of related interest: http://www.mathhelpforum.com/math-he...-function.html


    Another approach is to calculate F(u) = \Pr(Y < u) = \Pr(cX < u) = \Pr\left( X < \frac{u}{c}\right).

    Then the pdf of Y is f(u) = \frac{dF}{du}
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    Quote Originally Posted by 1of42 View Post
    [snip]
    2) If X1, X2, ..., Xn are independent and ~Normal(mu, sigma^2), show that (1/sigma^2)(X1^2 + X2^2 + ... + Xn^2)~gamma(n/2, 1/2).

    Any suggestions or help much appreciated.
    Here is the outline of one possible approach:

    Read this to get an idea of how to find the distribution of Y_i = \frac{1}{\sigma^2} \, X_i^2: http://www.mathhelpforum.com/math-he...tml#post119542

    Use this to get (or look up) the moment generating function of Y_i = \frac{1}{\sigma^2} \, X_i^2.

    Hence get the moment generating function of \sum_{i = 1}^{n} Y_i. Now identify this moment generating as being the same as that for the gamma distribution.


    Threads of related interest:

    http://www.mathhelpforum.com/math-he...a-samples.html

    http://www.mathhelpforum.com/math-he...tribution.html

    http://www.mathhelpforum.com/math-he...-function.html
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