(adsbygoogle = window.adsbygoogle || []).push({}); 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] ?

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# Conditional expectation, Lebesgue measure

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