could anyone please explain to me a simple question regarding expectations conditioned on sigma fields?
Consider a sample space {a, b, c} and $F_1$ = $\sigma$(a) and $F_2$ = $\sigma$(b) and a random variable X. X(a) = X(b) = 1 and X(c) = 0.
How do I calculate E(X|$F_1$) and E(E(X|$F_1$)|$F_2$) respectively and why? Thanks!
If $P$ is uniform, then $$E(X\mid F_1)=\frac12+\frac12\mathbf 1_{\{a\}},\qquad E(E(X\mid F_1)\mid F_2)=\frac34-\frac14\mathbf 1_{\{b\}}.$$