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Math Help - MGF

  1. #1
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    MGF

    The question is:
    Let X_{1}, X_{2}, ... X_{n} be independent and identically distributed exponential random variables each with mean \mu. Find the MGF of \Sigma_{i=1}^n X_{i} and verify that it is the MGF of a gamma random variable with parameters \alpha = n and \beta = \mu.

    The answer is:
    M_{\Sigma X_{i}}(\Theta) = \dfrac{1}{(1-\dfrac{\Theta}{\gamma})^{n}}, \Theta < \gamma

    I have the MGF for an exponential distribution and the MGF for a gamma distribution and the answer, but I cannot see how to put it all together. Any ideas?
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  2. #2
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    Re: MGF

    Quote Originally Posted by CountingPenguins View Post
    The question is:
    Let X_{1}, X_{2}, ... X_{n} be independent and identically distributed exponential random variables each with mean \mu. Find the MGF of \Sigma_{i=1}^n X_{i} and verify that it is the MGF of a gamma random variable with parameters \alpha = n and \beta = \mu.

    The answer is:
    M_{\Sigma X_{i}}(\Theta) = \dfrac{1}{(1-\dfrac{\Theta}{\gamma})^{n}}, \Theta < \gamma

    I have the MGF for an exponential distribution and the MGF for a gamma distribution and the answer, but I cannot see how to put it all together. Any ideas?
    You are expected to know what the mgf of a sum of independent random variables is. I suggest you review your class notes or textbook.
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