Poisson Process question (joint PMF and expectation)

Question

Stuck on this question, would really appreciate any help. Thanks!

Answer 1

Is it possible for $N$ to equal $3$ and $M$ to equal $2$? Why or why not?

The key to this question is to realize that if $L$ is defined as the difference $M-N$, then $N$ and $L$ are independent Poisson random variables because they are the numbers of arrivals in the (non-overlapping) intervals $(0,t]$ and $(t,t+s]$ respectively. (note that the problem statement has a a typo). Thus, for any $m \geq n \geq 0$, $$P\{N = n, M = m\} = P\{N = n, L = m-n\} = P\{N=n\}P\{L=m-n\}.$$

For part (b), note that $E[NM] = E[N(L+N)] = E[NL]+E[N^2]$.