- #1
spitz
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Homework Statement
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:
Homework 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]
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] ?