Binomial and Poisson Distributions

Question

$1$) If $X$, $Y$ are independent Binomial random variables with parameters $(n, p)$ and $(m, p)$, respectively, then $X+Y \sim Binomial(n+m, p)$.

$2$) If $X$,$Y$ are independent Poisson random variables with parameters $\lambda_1$ and $\lambda_2$,respectively, then $X+Y \sim Poisson (\lambda_1+\lambda_2)$.

How can I prove these with examples ??

Answer 1

Some hints:

Note that a $\text{Binomial}(n,p)$ random variable is the sum of $n$ independent $\text{Bernoulli}(p)$ random variables. From this fact, 1) should be easy to prove.

For 2), you have to get your hands dirty and compute the convolution $P(X+Y=k)=\sum_{i=0}^k P(X=i) P(Y=k-i)$. You can find this on this website and elsewhere.