Poisson arrivals happen in the given interval

Question

If $T_m$ is the $m^{\mathrm{th}}$ arrival time of a Poisson process, how to calculate $\mathbb P(T_m < a< T_{m+1} < b)$, where $a$ and $b$ are given times?

Answer 1

The event $T_m\lt a\lt T_{m+1}\lt b$ occurs exactly if there are exactly $m$ arrivals up to $a$ and then at least one arrival before $b$. If the Poisson process has rate $\lambda$, the probability for this is

$$ \frac{(\lambda a)^m\mathrm e^{-\lambda a}}{m!}\left(1-\mathrm e^{-\lambda (b-a)}\right)=\frac{(\lambda a)^m}{m!}\left(\mathrm e^{-\lambda a}-\mathrm e^{-\lambda b}\right)\;. $$