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Math Help - Poisson Variables and CLT

  1. #1
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    Poisson Variables and CLT

    I'm trying to show that for a Poisson random variable Y with mean \lambda, we get

    \displaystyle\frac{Y-\lambda}{\sqrt{\lambda}}\Rightarrow N

    as \lambda\rightarrow\infty

    If \lambda is a natural number n I'm okay, since Y is the sum of n independent Poisson variables with mean 1, and the CLT applies.

    My problem is extending to \lambda not necessarily a natural number. Any advice? Thanks!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Just let Y=\sum_{i=1}^{\lambda}X_i, where I was going to say what you thought.
    I assume you wanted X_i\sim P(1).

    It still is {Y-\mu_Y\over \sigma_Y} so it should be ok.

    Maybe use that same idea .....

    Let Y=\sum_{i=1}^{[\lambda]}X_i +X^*

    [\lambda] is the greatest integer less than or equal to lambda
    and X^* is an independent Poisson with mean \lambda -[\lambda]
    The CLT applies even if the sum doesn't consist of all iid rvs.
    And note that

    0\le {\lambda -[\lambda]\over\lambda}<{1\over\lambda}\to 0

    Just show the CLT applies to Y=\sum_{i=1}^{[\lambda]}X_i
    Last edited by matheagle; December 2nd 2009 at 11:35 PM.
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