Poisson probabilities:
Let Pn(t) be the probability that a Poisson process with parameter λ has n events in an interval of length t. Let h denote a very small interval of time (like ?t).
(a) Argue that
![](https://book.transtutors.com/qimg/79f4b404-2860-4bf2-9d0b-cce5150493b8.png)
(b) Substitute expressions for P1(h) and P0(h).
(c) Let h → 0 and derive a differential equation involving Pn(t), Pn(t), and Pn-1(t).
(d) Try a solution of the form Pn(t) = cnt ne-λt and evaluate cn.