Let $X_1$ be a random variable with poisson distribution $\text{Poisson}(\lambda_1)$ (i.e. $f(x)=\frac{\lambda^x}{x!}e^{-\lambda}$ if $x \in \{ 0,1,2,3,\ldots\}$ and $0$ otherwise) and let $X_2$ be a random variable with poisson distribution $\text{Poisson}(\lambda_2)$. What is the distribution of $Y = X_1 + X_2$?
I tried using the convolution formula:
$$f_y(Y) = \Sigma_{x=1}^{\infty} f_2(y-x)f_1(x) = \lambda_2^ye^{-(\lambda_1+\lambda_2)}\Sigma_{x=1}^{\infty}\frac{\left(\frac{\lambda_1}{\lambda_2}\right)^x}{x!(y-x)!}$$
But I am kind of stuck here..
Any ideas?
Thanks!