Thread: Conditional Expectation of Poisson Processes

Suppose that Xt is a Poisson Process with parameter λ = 1. Find E(X1 | X2) and E(X2 | X1).

I think they should be equal but I'm not sure how to write the value of the expectation of either.

Hey bjnovak.

Can you calculate the conditional distribution for P(X1=x|X2=y) and find its expectation?

You also have to tell us what kind of expectation you want to find since it is a multi-dimensional random variable: the expectation can be any general function of X1 and X2 or E[f(X1,X2)] and if you don't specify this then it won't make sense.

I believe the expectation is E[f(X1,X2)] where E( X2 | X1) (x) = f(X1,X2) is the (sum over X2 of X2*f(X1,X2) )
/ f over X1 (X1) (equation from book).

I'm not sure how the poisson process applies to this

That means that in this particular case f(X1,X2) = X2.

If you find the conditional distribution and get E[X2] of that distribution (it will be bivariate) then you can show what you need to show.