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.