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Math Help - Trouble understanding conditional expectation.

  1. #1
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    Trouble understanding conditional expectation.

    • "A Poisson number of Bernoulli trials." Fix λ > 0 and 0 < p < 1, and let N be Poisson(λ). Assume that the conditional distribution of Y, given that N = n, is binomial(n, p) for n = 0, 1, ..., Find the mean and variance of Y by conditioning on N.


    • "Conditional expectation given a sum." Fix n ≥ 2, let X1, . . . , Xn be n i.i.d. discrete random variables with finite expectation, and put Sn := X1 + X2 + Xn. Show that E[X1 | Sn] = Sn/n.

    I am having trouble understanding conditional expectation; the textbook I have doesn't do a very good job explaining it. Does anyone know of any resources online that can help me get a better grasp and understanding this?
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  2. #2
    MHF Contributor harish21's Avatar
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    Remember that both Poisson and Binomial are discrete distributions.

    Given: Y|N \sim \mbox{Binomial}(n,p)

    N \sim \mbox{Poisson}(\lambda)

    Use: E[E[Y|N]] = E[Y]

    or you can find the pmf of y and then the mean and variance.

    \displaystyle P(Y=y)=\sum_{n=0}^{\infty} f(y,n)

    \displaystyle  P(Y=y)=\sum_{n=0}^{\infty} f(Y|N=n) \cdot f(n)

    \displaystyle  P(Y=y)= \sum_{n=0}^{\infty} \dbinom{n}{y} p^y(1-p)^{n-y} \cdot \dfrac{e^{-\lambda} {\lambda}^n}{n!}

    ....
    Last edited by harish21; February 16th 2011 at 02:03 PM.
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