Counts and a first wait
A Poisson process has rate \lambda=3 per hour. Find the probability of exactly two events in 30 minutes, and the probability that the first wait exceeds 20 minutes.
Worked answer (click to show)
Thirty minutes is half an hour, so the mean count is 3/2. Therefore
\mathbb{P}\{N(1/2)=2\}=e^{-3/2}\frac{(3/2)^2}{2}.
Twenty minutes is 1/3 hour, so \mathbb{P}(T_1>1/3)=e^{-\lambda/3}=e^{-1}.