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Math Help - mgf of a sum of a poisson number of iid rvs

  1. #1
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    mgf of a sum of a poisson number of iid rvs

    Suppose that X_1,X_2,... are i.i.d. random variables, each of which has m.g.f. \psi(t). Let Y=X_1 + ... + X_N, where the number of terms N in the sum is a random variable having the Poisson distribution with mean \lambda. Assume that N and X_1,X_2,... are independent, and Y=0 if N=0. Determine the m.g.f of Y.


    I know the m.g.f. of Y|(N=n\ne0) is [\psi(t)]^n.
    I don't know where to go from there.
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  2. #2
    Grand Panjandrum
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    Re: mgf of a sum of a poisson number of iid rvs

    Quote Originally Posted by JJMC89 View Post
    Suppose that X_1,X_2,... are i.i.d. random variables, each of which has m.g.f. \psi(t). Let Y=X_1 + ... + X_N, where the number of terms N in the sum is a random variable having the Poisson distribution with mean \lambda. Assume that N and X_1,X_2,... are independent, and Y=0 if N=0. Determine the m.g.f of Y.


    I know the m.g.f. of Y|(N=n\ne0) is [\psi(t)]^n.
    I don't know where to go from there.
    \psi_Y(t)=E(e^{tY})=\sum_{n=0}^{\infty} p(n)E(e^{tY}|N=n)=\sum_{n=0}^{\infty} p(n)[\psi(t)]^n

    (the mgf when n=0 is still [\psi(t)]^n=1)

    CB
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