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Conditional expectation (discrete + continuous)

  1. Aug 29, 2006 #1
    I need help in solving the following problem:

    Let X be uniformly distributed over [0,1]. And for some c in (0,1), define Y = 1 if X>= c and Y = 0 if X < c. Find E[X|Y].

    My main problem is that I am having difficulty solving for f(X|Y) since X is continuous (uniform continuous over [0,1]) while Y is discrete (takes values of 0 or 1 only)
     
  2. jcsd
  3. Aug 29, 2006 #2

    HallsofIvy

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    Actually, "E[X|Y]" does not make sense to me. That is the expected value of X given that Y is what? E[X|Y= 1] can be done and E[X|Y= 0].

    Of course, if Y= 1, then X>= c so that is just E[X] where X is now uniformly distributed over [c, 1]. If Y= 0, then X is uniformly distributed over [0, c).
     
  4. Aug 29, 2006 #3
    The problem is asking for the best predictor of X based on Y which is essentially E(X|Y)...I believe I should have different answers for different values of Y.

    I got what you are saying though HallsofIvy. I'll try to work out the problem again. Thanks!
     
  5. Aug 30, 2006 #4
    Hi, an additional question regarding the problem...

    Given the question above, here's what I was able to get (that's suppose to be a chart below):

    X: ..........................[c,1]...................................[0,c)
    Y:.............................1........................................0
    E[X|Y]:...................(1+c)/2 ................................c/2
    X-E[X|Y]:..........[(c-1)/2, (1-c)/2]....................[-c/2, c/2 )

    Solving, I got Var(X) = (c^2)/12

    letting Z = X-E[X|Y]

    How do I solve for Var(Z)
     
    Last edited: Aug 30, 2006
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