I'm trying to understand some example of conditional expectation
So we have our probability space $ (\Omega, \mathcal{F}, P) $. Let $\Omega = \{1, 2, 3, 4\}, \mathcal{F} = 2^{|\Omega|}, P(w) = 1/4, w = 1,2, 3,4$. And let $G = \{ \varnothing, \{1,2\}, \{3, 4\}, \Omega \}- \sigma$-algebra and $G \subset \mathcal{F}$. And let random variable $X$ such that $X(w) =w^2$
So if I calculate the expectation of $X$, I'll get $$EX = \frac{1}{4}(1 + 4+9+16)$$
But when I calculate the conditional expectation $E[X|G]$ the result is somehow:
And I don't understand why is there $2$ and not $4$ in denominator. We are still in the same $w =1,2,3,4$ and the probability of a single element is $\frac{1}{|\Omega|}$.
Best Answer
We have $$E[X\mid\{1,2\}] = \frac{1^2\cdot P(\{1\})+2^2\cdot P(\{2\})}{P(\{1,2\})}=\frac{1\cdot\frac14+4\cdot\frac14}{\frac12}=\frac52,$$ and $$E[X\mid\{3,4\}]=\frac{3^2\cdot P(\{3\})+4^2\cdot P(\{4\})}{P(\{3,4\})}=\frac{9\cdot\frac14+16\cdot\frac14}{\frac12}=\frac{25}2.$$
The conditional expectation of $X$ given $G$ is $$E[X\mid G]=\mathbf{1}_{\{1,2\}}E[X\mid\{1,2\}]+\mathbf1_{\{3,4\}}E[X\mid\{3,4\}].$$