What is the distribution of an average of independent Poisson random variables? If I have random variables $X_1,X_2,\ldots,X_n$ that are Poisson distributed with parameters $λ_1,λ_2,\ldots,λ_n$, what is the distribution of $Y=1/n \sum X_i$? 
Who knows this can give an answer because I don't know how to do, please. 

If $X_1,\ldots,X_n$ are independent (something not mentioned in the question) then $X_1+\cdots+X_n\sim\mathrm{Poisson}(\lambda_1+\cdots+\lambda_n)$.  And then the distribution of $(X_1+\cdots+X_n)/n$ is a rescaling of that Poisson distribution, supported on the set $\left\{0, \dfrac 1 n, \dfrac 2 n, \dfrac 3 n, \ldots \right\}\vphantom{\dfrac{\displaystyle\sum}\sum}$.
