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Calculating the mean and variance from a moment generating function

  1. Aug 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume that X is squared-Chi-distributed, which means that the moment generating function is given by:

    [itex]m(t)=(1-2t)^{-k/2}[/itex]

    Use the mgf to find E(X) and var(X)

    3. The attempt at a solution
    I know that m'(0)=E(X), and m''(0)=var(X).

    So I find:

    [itex]m'(t)=k(1-2t)^{-(k/2)-1}[/itex]
    which gives m'(0)=k

    Similarily, I find

    [itex]m''(t)=(k^{2}+2k)(1-2t)^{-(k/2)-2}[/itex]
    which gives m''(0)=k^2+2k

    However, in my textbook, it says that the variance of a square-chi distribution should be 2k, not k^2. Where do I go wrong?
     
  2. jcsd
  3. Aug 22, 2012 #2

    LCKurtz

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    Your mistake is right there. m''(0) = E(X2), not var(X)

     
  4. Aug 22, 2012 #3
    Of course. Then var(X)=E(X^2)-(E(X))^2 =k^2+2k-k^2=2k.

    Thank you!!
     
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