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Thread: 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 $\displaystyle Y$ with mean $\displaystyle \lambda$, we get

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

    as $\displaystyle \lambda\rightarrow\infty$

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

    My problem is extending to $\displaystyle \lambda$ 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 $\displaystyle Y=\sum_{i=1}^{\lambda}X_i$, where I was going to say what you thought.
    I assume you wanted $\displaystyle X_i\sim P(1)$.

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

    Maybe use that same idea .....

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

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

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

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