Let pnt be the probability that a poisson process with

Question

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

(b) Substitute expressions for P₁(h) and P₀(h).

(c) Let h → 0 and derive a differential equation involving P_n(t), P_n(t), and P_n-1(t).

(d) Try a solution of the form P_n(t) = cnt ne-λt and evaluate c_n.