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Let $X$ have the Gamma$(s,1)$ and given $X=x$, let $Y$ have the Possion distribution with parameter $x$. Show that $$\frac{Y-E(Y)}{\sqrt{\operatorname{var}(Y)}}\longrightarrow W$$ where $\longrightarrow$ means converges in distribution as $s$ goes to infinity. And $W$ needs to be identified.


I have worked out the moment generating function of $Y$, $$ M_Y(t)=\left(\frac{1}{2-e^{t}}\right)^s$$ Then I work out the mgf of $\frac{Y-E(Y)}{\sqrt{\operatorname{var}(Y)}}$, $$ M(t)=e^{-\frac{s}{\sqrt{2s}}t}\left(\frac{1}{2-e^{\frac{t}{\sqrt{2s}}}}\right)^s$$ But I don't know what does it converges to.

Anything wrong with my above calculation?

Thanks.

  • 1
    [take a look at that](http://math.stackexchange.com/questions/87061/help-with-convergence-in-distribution/87067#87067)2011-12-09
  • 0
    You said "with parameter $x$". Did you actually mean "with parameter $X$"? That would at least make the question make sesne.2011-12-09
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    @MichaelHardy I think they are the same, since my question is given X=x and Y hase Poisson with parameter x.2011-12-09

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