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

$X(w)=5w^2$

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

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

$X(w)=5w^2$

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

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