Distribution of arrival times of Poisson point processes

Question

Let $(M_{t})_{t\geq 0}$ and $(N_t)_{t\geq0}$ be two independent Poisson point processes with rate $\lambda$ and $\mu$ respectively. Let $\tau$ be the first arrival time for the process $N_{t}$. Find:

a) the distribution of $M_{\tau }$

b) the distribution of $N_{2\tau} - N_{\tau}$

I was hoping for some clarification as to how the distribution of $M_\tau$ changes based on the arrival time of $N_t$, as I just don't understand how it could if the processes are independent?

Thanks in advance

Answer 1

Hints: $P(M_{\tau}=k)=\sum_j P(\tau =j, M_j=k)$ and the events $(\tau =j), (M_j=k)$ are independent. Can you compute this now?

For the second part $P(N_{2\tau}-N_{\tau} =k)=\sum_j P(\tau =j,N_{2j}-N_{j} =k)$. Use the fact that $(\tau =j)$ and $(N_{2j}-N_{j} =k)$ are independent (because poissson process has independent increments).