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Math Help - Conditional Expectation involving Poisson Distribution

  1. #1
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    Conditional Expectation involving Poisson Distribution

    Let X1,...,Xn be a random sample from a Poisson distribution with mean theta. Find E(Y_1 = X_1 + 2X_2 + 3X_3 | Y_2 = \Sigma X_i)

    I know that \Sigma X_i = Pois(\theta n)

    I need to find P(Y_1|Y_2) = \frac{P(Y_1,Y_2)}{P(Y_2)}

    How would I find the joint PMF of Y1 and Y2?
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  2. #2
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    Hello,

    I think the formula you use is only available with probability distributions with densities.

    Anyway, look here : http://www.mathhelpforum.com/math-he...757-post2.html

    which proves that E[X_i|Y_2]=\frac{Y_2}{3}~,~\forall i \in\{1,2,3\}

    And then just write E[Y_1|Y_2]=E[X_1|Y_2]+2E[X_2|Y_2]+3E[X_3|Y_2] (linearity of the expectation)

    And you're done.
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