1. ## Conditional Expectation

SOLVED

Consider a Poisson process with parameter $\displaystyle \lambda$ . Let $\displaystyle X$ be the number of events in $\displaystyle (0,3]$ and $\displaystyle Y$
the number of events in $\displaystyle (2,4]$.
(a) Find the mean and variance of $\displaystyle Y-X$.
(b) Find $\displaystyle E(Y | X)$. Verify that $\displaystyle E[E(Y | X)] = E(Y)$.

I've done (a), and got the answers $\displaystyle -\lambda$ and $\displaystyle 3\lambda$ 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 $\displaystyle f_{Y|X} (y|x)$ (more specifically, I'm not able to calculate $\displaystyle P(X=x, Y=y)$) since the two variables are not independent.

EDIT: I've made some progress in calculating $\displaystyle P(X=x, Y=y)$:

It's easy if $\displaystyle x,y$ were actually known integers, but since they're not I figured that we could let the number of events in $\displaystyle (2,3]$ be denoted by $\displaystyle a$. Hence we get:

$\displaystyle P(X=x, Y=y) = \sum_{a=0}^{\min(x,y)} (\frac{\lambda^{x-a}}{(x-a)!}e^{-2\lambda})(\frac{\lambda^a}{a!}e^{-\lambda})(\frac{\lambda^{y-a}}{(y-a)!}e^{-\lambda})$

..and I'm stuck.

SOLVED