Poisson process Probabilities

Question

If I assume that $\{N(t)=: t \ge 0\}$ is a Poisson process with intensity $\lambda$. For $0<s<t$, how would I find the $\Pr\{N(t)>N(s)\}$?

Answer 1

Look at $P(N(t)-N(s)>0) \sim \mathrm{Poisson}(\lambda(t-s))$ which is related to the independent increments property.

if $M \sim \mathrm{Poisson}(\lambda(t-s))$, $P(M>0) = 1-P(M=0) = 1-e^{\lambda(t-s)}$