Conditional expectation, Lebesgue measure

[spitz]

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

X(w)=5w^2

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

The Attempt at a Solution

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

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

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

or P(A_1)=\frac{1}{4}, and P(A_2)=\frac{3}{4} ?

 

[Ray Vickson]

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

X(w)=5w^2

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

The Attempt at a Solution

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

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

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

or P(A_1)=\frac{1}{4}, and P(A_2)=\frac{3}{4} ?

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

RGV

 

[spitz]

oh. 1/4