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Thread: Poisson Sampling

  1. #1
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    Poisson Sampling

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

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

    Then insert the n, since $\displaystyle Y_n=S_n/n$
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