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)

thank you

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- June 21st 2011, 07:35 AMalehandr12Poisson process question
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)

thank you - June 22nd 2011, 04:48 AMSpringFan25Re: Poisson process question
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:

Where:

you can subsitute the formulae for the PMFs to finish. Dont forget that the probability is zero if m > k