Question

Show that If a counting process {N(t) : t ≥ 0} is a Poisson process with rate λ, then P{N(s + t) − N(s) = n} = e −λt((λt)^ n/ n!) .

Answer 1

A Poisson process with rate (or intensity) λ > 0 is a counting process N(t) such that

1. N(0) = 0;

2) poisson process has independent increments:

3) number of events in any interval of length t is Poisson(λt)

hence

N(s+t) - N(s) follow poisson (λt)

P{N(s + t) − N(s) = n}   = e ^−λt((λt)^ n/ n!) .