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.