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Math Help - Conditional Expectation

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    Conditional Expectation

    SOLVED

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

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




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

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

    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
    Last edited by h2osprey; February 4th 2010 at 04:18 PM.
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