Find the distribution of numbers of arrivals of the Poisson process $N(t)$ in time interval $[t, t+\tau)$, $\tau \sim Exp(a)$.

Question

Poisson process has rate $\lambda$ and $\tau \perp \!\!\! \perp N(t)$. To find distribution i've started with $P(N(t+\tau)-N(t)=k) = P(N(\tau) = k)$. I know that $N(t) \sim Poiss(\lambda t)$, but i don't know what to do next to find distribution.

Answer 1

Hint: use the formula $$ P(N(\tau) = k) = \int_0^\infty P(N(s)=k|\tau = s) ae^{-as}\, ds.$$