
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 $\displaystyle YX$.
(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 $\displaystyle f_{YX} (yx)$ (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:
$\displaystyle P(X=x, Y=y) = \sum_{a=0}^{\min(x,y)} (\frac{\lambda^{xa}}{(xa)!}e^{2\lambda})(\frac{\lambda^a}{a!}e^{\lambda})(\frac{\lambda^{ya}}{(ya)!}e^{\lambda})$
..and I'm stuck.
SOLVED