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Conditional expectation, Lebesgue measure

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Let Ω = [0,1] with the σ-field of Borel sets and let P be the Lebesgue measure on [0,1]. Find E(X|Y) if:

    2. Relevant equations

    [itex]X(w)=5w^2[/itex]

    [itex]Y(w)= \left\{ \begin{array}{ll}
    4 & \mbox{if $w \in [0,\frac{1}{4}]$} \\
    2 & \mbox{if $w \in (\frac{1}{4},1]$} \\
    \end{array}
    \right.[/itex]

    3. The attempt at a solution

    For [itex]w\in A_1=[0,\frac{1}{4}][/itex]:

    [itex]E(X|Y)(w)=E(X|A_1)=\frac{\int_{A_1}x\,dp}{P(A_1)}=\frac{1}{{P(A_1)}} \displaystyle\int_{0}^{1/4}5w^2\,dw[/itex]

    Do I use [itex]P(A_1)=P(A_2)=\frac{1}{2}[/itex],

    or [itex]P(A_1)=\frac{1}{4}[/itex], and [itex]P(A_2)=\frac{3}{4}[/itex] ?
     
  2. jcsd
  3. Feb 22, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    You said that P was Lebesgue measure, so what do you think is the Lebesgue measure of [0,1/4]?

    RGV
     
  4. Feb 22, 2012 #3
    oh. 1/4
     
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