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Probability theory 2

  1. Feb 6, 2008 #1
    1. The problem statement, all variables and given/known data
    The homework question I have is:

    Let X be uniform over (0,1). Find E[X|X<1/2].


    3. The attempt at a solution

    First I found the density function: f(x|x<1/2] = f(x∩x<1/2) / f(x<1/2) = f(x<1/2) / f(x<1/2) = 1.

    So, E[X|X<1/2] = ∫ xdx


    My question is... are the limits of integration suppose to be from 0 to 1, or 0 to 1/2? Is everything else correct?

    Thanks in advance
     
  2. jcsd
  3. Feb 6, 2008 #2

    EnumaElish

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    The limits of integration are {0, 1/2} because f(x|X<1/2) is defined (or is positive) for x<1/2 only.

    This information should also alert you to the fact that [tex]\int_0^{1/2}f(x|X<1/2)dx = 1[/tex] has to be the case. Does f(x|x<1/2) = 1 satisfy this condition?
     
  4. Feb 6, 2008 #3
    Okay, new strategy. In my textbook in a similar problem, they did:

    f(x|x<1/2) = f(x) / P{x<1/2}
    When I do that, I get
    f(x) / P{x<1/2} = 1 / (1/2) = 2.

    E[x|x<1/2] = [tex]\int_0^{1/2}2xdx [/tex]
    = 1/4
     
    Last edited: Feb 6, 2008
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