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Math Help - Find mu (mean); E [X (X - 1)] and sigma squared (variance) for each of the following

  1. #1
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    Exclamation Find mu (mean); E [X (X - 1)] and sigma squared (variance) for each of the following

    Find mu (mean); E [X (X - 1)] and sigma squared (variance) for each of the following distributions:
    1. f (x) = (1 / x! ) (1/2)^5 (5! / (5 - x)! ) ; for x = 0, 1, 2, 3, 4, 5
    2. f (x) = (e^ -a) (a^x)/(x!) for x = 0, 1, 2,... (a > 0)


    any help will be very much appreciated I'm so lost on this homework question.
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  2. #2
    MHF Contributor matheagle's Avatar
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    I'll do 2, since I mentioned that in class last week

    Here you have a Poisson and your mean is a.

    You calculate the mean via the Taylor Series for e^a=\sum_{x=0}^{\infty} {a^x\over x!}

    so \mu =\sum_{x=0}^{\infty} {xe^{-a}a^x\over x!} =ae^{-a}\sum_{x=1}^{\infty} {a^{x-1}\over (x-1)!}

    Let y=x-1 and you have

    =ae^{-a}\sum_{y=0}^{\infty} {a^y\over y!}= ae^{-a}e^a=a

    IN ORDER to get the variance NOTE that we need the second moment

    BUT it's easier to get E(X(X-1))=E(X^2)-E(X)=E(X^2)-a which is NOT the variance.

    NOW do the same thing I did above with this and it's over...

    E(X(X-1))=\sum_{x=0}^{\infty} {x(x-1)e^{-a}a^x\over x!}=a^2e^{-a}\sum_{x=2}^{\infty} {a^{x-2}\over (x-2)!}

    NOW let y=x-2 and you have

    =a^2e^{-a}\sum_{y=0}^{\infty} {a^y\over y!}= a^2e^{-a}e^a=a^2

    Add this up and the variance is a.
    Last edited by matheagle; September 27th 2009 at 07:35 PM.
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    Thank you.
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    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Intsecxtanx View Post
    Thank you for the advice but our professor hasn't discussed the poisson distribution and advised us not to go that route.
    I have no idea what other route to take.

    The idea is to solve for the second moment via E(X(X-1))
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    Smile

    I'm so sorry, I edited my last comment because I didn't realize how thorough your explanation was and how helpful it actually was. I really, really appreciate your help and will use this knowledge to complete my homework! Thank you
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