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Math Help - Probability help please

  1. #1
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    Probability help please

    If X has generating function G, then the expectation(X) = G'(1) and variance(X) = G''(1) + G'(1) - G'(1)^2.

    Use this to prove that if X and Y are independent, then variance(X+Y) = variance(X) + variance(Y).
    Last edited by clockingly; October 26th 2007 at 07:40 PM.
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  2. #2
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    Quote Originally Posted by clockingly View Post
    If X has generating function G, then the expectation(X) = G'(1) and variance(X) = G''(1) + G'(1) - G'(1)^2.

    Use this to prove that if X and Y are independent, then variance(X+Y) = variance(X) + variance(Y).
    Let X have generating function G, and Y have generating function H.

    Then as X and Y are independent the generating function of X+Y is:

     F(z)=H(z)G(z).

    and as these are probability generating functions H(1)=G(1)=1.

    You should now be able to complete this yourself, if not let us know and
    we will finnish this off for you.

    RonL
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  3. #3
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    Thanks - but I'm still a little confused. I get that H(1) = G(1) = 1, but I'm not sure how to relate that to the variance of X and Y.
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  4. #4
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    Quote Originally Posted by clockingly View Post
    Thanks - but I'm still a little confused. I get that H(1) = G(1) = 1, but I'm not sure how to relate that to the variance of X and Y.
    variance(X+Y) = F''(1) + F'(1) - F'(1)^2

    Now use the product rule repeatedly on F(z)=H(z)G(z) to get F'(z) and F''(z)
    in terms of H''(z), H'(z), H(z), G''(z), G'(z) an G(z). Now put z=1 and
    substitute into the equation for the variance of X+Y and the result drops out.

    RonL
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  5. #5
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    okay so

    F'(z) is H(z)G'(z) + G(z)H'(z).

    F''(z) is 2*(H'(z)G'(z)) + H(z)G"(z) + G(z)H"(z)

    so variance(X+Y) = F"(1)+ F'(1) - F'(1)^2

    so it is the same as:

    2*(H'(1)G'(1)) + H(1)G"(1) + G(1)H"(1) +

    H(1)G'(1) + G(1)H'(1) +

    (H(1)G'(1) + G(1)H'(1))^2

    however, how do i reduce this?
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  6. #6
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    Quote Originally Posted by clockingly View Post
    okay so

    F'(z) is H(z)G'(z) + G(z)H'(z).

    F''(z) is 2*(H'(z)G'(z)) + H(z)G"(z) + G(z)H"(z)

    so variance(X+Y) = F"(1)+ F'(1) - F'(1)^2

    so it is the same as:

    2*(H'(1)G'(1)) + H(1)G"(1) + G(1)H"(1) +

    H(1)G'(1) + G(1)H'(1) +

    (H(1)G'(1) + G(1)H'(1))^2

    however, how do i reduce this?
    First replace all the G(1)'s and H(1)'s by 1, so:


    2*(H'(1)G'(1)) + H(1)G"(1) + G(1)H"(1) + H(1)G'(1) + G(1)H'(1) - (H(1)G'(1) + G(1)H'(1))^2


    .... = 2*(H'(1)G'(1)) + G"(1) + H"(1) + G'(1) + H'(1) - (G'(1) + H'(1))^2

    .... = 2*(H'(1)G'(1)) + G"(1) + H"(1) + G'(1) + H'(1) - (G'(1))^2 - (H'(1))^2 -2*(H'(1)G'(1))

    .... = G"(1) + H"(1) + G'(1) + H'(1) - (G'(1))^2 - (H'(1))^2

    .... = G"(1) + G'(1) - (G'(1))^2 + H"(1) + H'(1) - (H'(1))^2

    .... = var(X) + var(Y)

    RonL
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