Given a poisson process Z(t) with a given rate lamda, k and m nonnegative integers and t and c real and positive numbers, calculate the probability:
P(Z(t-c)=m | Z(t)=k)
This is a poisson process so we have the following properties:
Start from the usual formula for conditional probabilities
The denominator is easy using property [B].
The numerator can be re-written as follows:
using property [C], the events and are independent, and hence
Both of those terms follow a poisson distribution. (unless k-m is negative, then the probability is zero).
So the final answer is:
you can subsitute the formulae for the PMFs to finish. Dont forget that the probability is zero if m > k