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Conditional expectation on multiple variables

  1. May 20, 2010 #1
    How to compute [tex]E[X|Y1,Y2][/tex]?
    Assume all random variables are discrete.

    I tried [tex]E[X|Y1,Y2] = \sum_x{x p(x|y1,y2)[/tex] but I'm not sure how to compute [tex]p(x|y1,y2] = \frac{p(x \cap y1 \cap y2)}{p(y1 \cap y2)}[/tex]

    If it is correct, how can I simplify the expression if Y1 and Y2 are iid?
     
  2. jcsd
  3. May 20, 2010 #2

    EnumaElish

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    If y1 and y2 are independent then p(y1, y2) = p(y1)p(y2).
     
  4. May 20, 2010 #3

    statdad

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    In general

    [tex]
    p(x \mid y_1, y_2) = \frac{p(x,y_1,y_2)}{p(y_1,y_2)}
    [/tex]

    where the numerator is the joint density (or mass function for discrete case) of all three, and the denominator is the marginal of the two ys. You treat this as a function of [itex] x [/itex] alone. Then, in the discrete case, the expected value is

    [tex]
    \sum x p(x \mid y_1, y_2)
    [/tex]

    and in the continuous case it is

    [tex]
    \int x p(x \mid y_1, y_2) \, dx
    [/tex]

    In each case it is possible for the answer to depend on both [itex] y_1, y_2 [/itex].
     
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