SOLVED

Consider a Poisson process with parameter . Let be the number of events in and

the number of events in .

(a) Find the mean and variance of .

(b) Find . Verify that .

I've done (a), and got the answers and for mean and variance respectively.

I can't figure out (b) - I know the formula for calculating conditional expectation, but can't seem to figure out how to find (more specifically, I'm not able to calculate ) since the two variables are not independent.

EDIT: I've made some progress in calculating :

It's easy if were actually known integers, but since they're not I figured that we could let the number of events in be denoted by . Hence we get:

..and I'm stuck.

SOLVED