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Math Help - calculating MGFs

  1. #1
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    calculating MGFs

    An insurance company receives i.i.d. claims X_{1}, X_{2},...X_{N} where N ~ Poisson(\lambda). Let X=X_{1}+X_{2}+...X_{N}. Suppose E(X_{i})=\mu and Var(X_{i})=\sigma^2

    a) Find E(X), var(X): I got this part with E(X)= \lambda \mu and var(X)= \lambda ^2 \sigma ^2 + \mu ^2 \lambda


    b) Suppose that Xi is Gamma(\alpha , \beta). Find the MGF of X:
    I tried MGF(t)= (\frac{\beta}{\beta -t})^{\alpha\lambda} because E(N)=\lambda and the sum of independent variables' MGF is just the product of each MGF, but the results doesn't match part (a). Please help me out with this part. Thanks!
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  2. #2
    MHF Contributor harish21's Avatar
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    can you show your work for part (a)?

    (b)for independent random variables:

    E[e^{tX}]= E[e^{tX_1}\cdot e^{tX_2}\cdots e^{tX_n}]= E[e^{tX_1}]\cdot E[e^{tX_2}]\cdots E[e^{tX_n}]

    where X_1, X_2,...X_n are random variables following a gamma distribution.
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  3. #3
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    Quote Originally Posted by harish21 View Post
    can you show your work for part (a)?

    (b)for independent random variables:

    E[e^{tX}]= E[e^{tX_1}\cdot e^{tX_2}\cdots e^{tX_n}]= E[e^{tX_1}]\cdot E[e^{tX_2}]\cdots E[e^{tX_n}]

    where X_1, X_2,...X_n are random variables following a gamma distribution.
    For E(X), I think it's easy to see.
    Var(X)=E(Var(X|N))+Var(E(X|N))=E(N^2 \sigma ^ 2)+Var(N \mu) = \lambda  \sigma^2 + \mu^2 \lambda

    EDIT: now that I think about it, should it be \lambda \sigma^2 + \mu^2 \lambda ?
    Last edited by brogers; February 7th 2011 at 05:47 PM. Reason: a mistake?
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  4. #4
    MHF Contributor matheagle's Avatar
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    I believe you need E_N\bigl((1-\beta t)^{-\alpha N}\bigr)

    Try the substitution e^s= (1-\beta t)^{-\alpha}

    So you have to figure out E(e^{sN})=e^{-\lambda (e^s-1)}=e^{-\lambda[(1-\beta t)^{-\alpha}-1]}
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  5. #5
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    Quote Originally Posted by brogers View Post
    An insurance company receives i.i.d. claims X_{1}, X_{2},...X_{N} where N ~ Poisson(\lambda). Let X=X_{1}+X_{2}+...X_{N}. Suppose E(X_{i})=\mu and Var(X_{i})=\sigma^2

    [snip]

    b) Suppose that Xi is Gamma(\alpha , \beta). Find the MGF of X:
    I tried MGF(t)= (\frac{\beta}{\beta -t})^{\alpha\lambda} because E(N)=\lambda and the sum of independent variables' MGF is just the product of each MGF, but the results doesn't match part (a). Please help me out with this part. Thanks!
    pp17-18 here have the formula you need to use: http://www.markirwin.net/stat110/Lecture/Section45.pdf
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  6. #6
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    Got it!! The pdf really helped. Thanks!
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