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Math Help - Limiting Distribution of Poisson RV

  1. #1
    Pur
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    Limiting Distribution of Poisson RV

    Let Z_n\sim poisson(\mu=n). Show that the limiting distribution of Y_n=\frac{Z_n-n}{\sqrt{n}} is a standard normal distribution.

    If anyone can point me in the right direction it would be appreciated. So far I have already attempted to use the mgf method:
    \mathbb{E}(\exp(tY_n))= \exp\left\{\frac{-tn}{\sqrt{n}}\right\}\mathbb{E}\left(\frac{tZ_n}{\  sqrt{n}} \right)
    and using the fact that \mathbb{E}\left( tZ_n\right)=e^{n(e^{t}-1)}, we have \exp\left\{\frac{-tn}{\sqrt{n}}\right\}e^{n(e^{t/\sqrt{n}}-1)} .However, I do not see how the limiting distribution of this will converge to the mgf for a standard normal.
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  2. #2
    MHF Contributor

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    You could say that the Poisson distribution of parameter n is the distribution of the sum of n i.i.d. Poisson(1) random variables, and therefore simply apply the CLT.

    Or you compute the limit of the mgf: the exponent in the mgf is -t\sqrt{n}+n(e^{t/\sqrt{n}}-1)=-t\sqrt{n}+n(\frac{t}{\sqrt{n}}+\frac{t^2}{2n}+o(\f  rac{1}{n})) =\frac{t^2}{2}+o(1), i.e. its limit is \frac{t^2}{2} therefore you get that the limit of the mgf is e^{t^2/2}.
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