Sum of independent random variables with different distributions

Question

Can we find the distribution of the sum of random variables with different pmf and different possible values?For example let X be a Poisson random variable with parameter $\lambda$ and Y an independent Bernoulli random variable with parameter $p$. Then is the probability mass function of X + Y $$p_{X+Y}(n)=\sum_{k=1}^n P(X=n-k)P(Y=k) = e^{-\lambda}\frac{\lambda^n}{n!}(1-p)+e^{-\lambda}\frac{\lambda^{n-1}}{(n-1)!}p = pe^{-\lambda}(\frac{\lambda^{n-1}n-\lambda^n}{n!})+e^{-\lambda}\frac{\lambda^{n}}{n!}?$$

Answer 1

if $n=0$, then we have $$p_{{X+Y}}(0)=\exp(-\lambda)(1-p)$$

if $n>0$, then we have

\begin{align} p_{X+Y}(n) &= \sum_{k=0}^1 P(X=n-k) P(Y=k)\\ &= e^{-\lambda}\frac{\lambda^n}{n!}(1-p)+e^{-\lambda}\frac{\lambda^{n-1}}{(n-1)!}p \\&= pe^{-\lambda}(\frac{\lambda^{n-1}n-\lambda^n}{n!})+e^{-\lambda}\frac{\lambda^{n}}{n!} \end{align}