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Math Help - Poisson Sampling

  1. #1
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    Poisson Sampling

    Hi, here's the question:
    Let X_1,X_2,...,X_n be independent r.v.s with the Poisson distribution of parameter m > 0.
    Let Y = (X_1 + ... + X_n)/n. Find the support and probability mass function of the distribution of Y. Calculate E[Y-m] and E[(Y-m)^2].
    Any help would be appreciated. Thanks.
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  2. #2
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    I get E [Y] = E [1/n(X_1 + ... + X_n)] = (1/n) E [(X_1 + ... + X_n)]= (1/n) nm = m using linearity. Also, Var[Y] = m/n. But this is no longer a Poisson since the mean and variance are different?? If that's true then E[Y-m]=0. Is this right?
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  3. #3
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    Quote Originally Posted by davidmccormick View Post
    I get E [Y] = E [1/n(X_1 + ... + X_n)] = (1/n) E [(X_1 + ... + X_n)]= (1/n) nm = m using linearity. Also, Var[Y] = m/n. But this is no longer a Poisson since the mean and variance are different?? If that's true then E[Y-m]=0. Is this right?
    Indeed Y is no longer a Poisson r.v., but you could have seen that directly from the question about the support of Y. While X_1,\ldots,X_n are integers, their mean Y may not be an integer. The support of the distribution of Y is "the set of possible values" for Y (You can remove the " " in the present situation). Therefore, it is...

    And in order to determine the probability mass function of Y, you should first find that (or recall that) of X_1+\cdots+X_n. It is Poisson with parameter nm. Then giving the probability mass function of Y should be simple. Tell us if you still have problems.

    (and yes of course E[Y-m]=0)
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  4. #4
    MHF Contributor matheagle's Avatar
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    Just let S_n=X_1+\cdots+X_n and find it's distribution.

    Then insert the n, since Y_n=S_n/n
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