Probability, poisson distribution

Question

Can anyone help with the following question?

Let $X_1,\ldots,X_n$ be independent Poisson random variables with parameter $\lambda$. What is the mass function of $X_1+\ldots+X_n$?

Answer 1

\begin{align} P(X_1+ X_2 =k) &= \sum_{i = 0}^k P(X_1+ X_2 = k, X_1 = i)\\ &= \sum_{i=0}^k P(X_2 = k-i , X_1 =i)\\ &= \sum_{i=0}^k P(X_2 = k-i)P(X_1=i)\\ &= \sum_{i=0}^k e^{-\lambda}\frac{\lambda^{k-i}}{(k-i)!}e^{-\lambda}\frac{\lambda^i}{i!}\\ &= e^{-2\lambda}\frac 1{k!}\sum_{i=0}^k \frac{k!}{i!(k-i)!}\lambda^{k-i}\lambda^i\\ &= \frac{(2\lambda)^k}{k!}e^{-2\lambda} \end{align}

Now let $S_n=\sum_{j=1}^n{X_j}$. We can easily prove that $S_n\sim\operatorname{Po}(n\lambda)$ using a similar approach.