Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

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


    User Avatar
    Homework Helper

    In general

    p(x \mid y_1, y_2) = \frac{p(x,y_1,y_2)}{p(y_1,y_2)}

    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

    \sum x p(x \mid y_1, y_2)

    and in the continuous case it is

    \int x p(x \mid y_1, y_2) \, dx

    In each case it is possible for the answer to depend on both [itex] y_1, y_2 [/itex].
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook