Exact distribution of mean of Poisson random variables

Question

Let $Y_1,\dots,Y_n$ be independent Poisson distributed random variables with parameter $ \lambda$. Consider $ \bar{Y}_n $. I know that the mean is an asymptotic normal random variable. How can I find the exact distribution hereof?

Thanks.

Answer 1

Because the $Y_i$ are i.i.d Poisson it follows that $\sum_{i=1}^n Y_i\sim \text{Poi}(n\lambda)$. In particular for $k\in\{0,1/n,2/n,\dotsc,\}$ we have that $$ P(\bar{Y}_{n}=k)=P\left(\sum_{i=1}^nY_i=nk\right)=\frac{(n\lambda)^{nk}}{(nk)!} e^{-n\lambda}$$