How does one compute conditional expectation with respect to a continuous random variable

conditional probabilityconditional-expectationmeasure-theoryprobability theoryrandom variables

I can't wrap my head around the way to compute conditional expectation with respect to a continuous random variable. For instance, consider a probability space $(\Omega, A, P)$, where $\Omega = [0,1]$ and $A$ is a corresponding Borel $\sigma$-algebra. Let's define random variables $X(\omega)=\omega$ and $Y=\sin (\pi \omega)$. How does one compute the expression for $\mathbb{E}(X\mid \mathcal{B})$, where $\mathcal{B}$ is the $\sigma$-algebra generated by $Y$?

If $Y$ was discrete, such as
$$
Y = \begin{cases}
1,&\omega\in[0,1/2]\\
0,&\omega\in(1/2,1]
\end{cases}
,$$

then I understand that $\mathbb{E}(X\mid \mathcal{B})$ would be equal to
$$
\mathbb{E}(X\mid \mathcal{B}) = \frac{1}{4}\mathbb{I}_{Y=1}+\frac{1}{4}\mathbb{I}_{Y=0}.
$$

However, in the continuous case I'm not so sure. My intuition suggests that it's something like $\sin^{-1}(y)/\pi$; however, then
$$
\mathbb{E}(\mathbb{E}(X\mid Y)) = \int_Y \mathbb{E}(X\mid Y)d\omega = \frac{2}{\pi}\cdot\int_0^1 \arcsin(y) dy \neq \mathbb{E}(X).
$$

Best Answer

For instance, consider a probability space $(Ω,\mathcal A,P)$, where $Ω=[0,1]$ and $A$ is a corresponding Borel σ-algebra. Let's define random variables $X(ω)=ω$ and $Y=sin(πω)$. How does one compute the expression for $E(X∣\mathcal B)$, where $\mathcal B$ is the σ-algebra generated by Y?

Using $\arcsin: [0..1]\mapsto [0..\pi/2]$

A null events described by $\{\omega\in\Omega:Y(\omega)=y\}$, (where $y\in[0..1)$ ) will contain exactly two outcomes with no bias: $$\{\omega\in\Omega:Y(\omega)=y\}=\{\arcsin(y)/\pi, 1-\arcsin(y)/\pi\}$$ Therefore $\forall y\in Y(\Omega)~,\mathsf E(X\mid Y=y)=1/2$, so since this holds for any $y$, ...$$\mathsf E(X\mid\mathcal B)=(1/2)\mathbf 1_{Y\in[0..1]}$$

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