So I am just learning about conditional expectation in the modern probability sense. I understand that it is a random variable.
What I am having a hard time understanding is how you calculate the outputs of the conditional expectation in specific examples.
For instance I'm reading through this example and the author doesn't explain how they get their values for the conditional expectation:
Let $\Omega=\{a,b,c\}, \ \mathcal{F}_1=\sigma(\{a\}), \ \mathcal{F}_2=\sigma(\{c\}).$ Take $X(b)=1, \ X(a)=X(c)=0.$ In this case we have $$ E(X|\mathcal{F}_1)(a)=0, \ E(X|\mathcal{F}_1)(b)=\frac{1}{2}, \ E(X|\mathcal{F}_1)(c)=\frac{1}{2}. $$
Could somebody please explain how they arrived at these values?
Thanks!
The $\sigma$-field ${\cal F_1}$ is given by $\{\{a\}, \{b,c\}, \emptyset , \Omega \}$. The definition is that a version of $E[X|{\cal F_1}]$ is a ${\cal F_1}$ measurable random variable $Y$ such that $\int_A Y = \int_A X$ for any $A \in {\cal F_1}$.
$Y$ must be ${\cal F_1}$ measurable. In particular, this means it must take the same value at $b$ and $c$ (otherwise it would not be measurable). Hence it has the form $Y = Y_1 1_{ \{a\} } + Y_2 1_{ \{b,c\} }$, and all we need to do is to compute values for $Y_1,Y_2$ that satisfy the defining equation.
Choosing $A = \{a\}$ shows that $ \int_A Y= Y_1 p \{a\} = X(a) p \{a\}= 0$, hence if $p\{a\} > 0$ we have $Y_1 = X(a)$ and if $p \{a\} = 0$, the value doesn't matter (we only need the values to be defined ae.), so we can choose $Y_1 = X(a)$.
Choosing $A= \{b,c\}$ gives $ \int_A Y= Y_2 p \{b,c\} = \int_A X = X(c) p \{c\} + X(b) p \{b\}$. As above, if $p\{b,c\} >0$ we have $Y_2 = {X(c) p \{c\} + X(b) p \{b\} \over p \{b,c\} }$, and if $p\{b,c\} =0$, we can pick any value.