conditional expectation of squares

Question

Say we want $E_{I}(E(X|I))^2$ where $X$ is a continuous random variable and $I=0$ or $1$ is a discrete random variable. Can we write the following ?

$E_{I}(E(X|I))^2 = (E(X|I=1))^{2}\times P(I=1) + (E(X|I=0))^{2}\times P(I=0)$

Answer 1

Your use of parentheses is somewhat ambiguous. To be clear: $$\begin{align*} {\rm E}[({\rm E}[X \mid I])^2] &= ({\rm E}[X \mid I = 1])^2 \Pr[I = 1] + ({\rm E}[X \mid I = 0])^2 \Pr[I = 0], \\ ({\rm E}[{\rm E}[X \mid I]])^2 &= \left( {\rm E}[X \mid I = 1] \Pr[I = 1] + {\rm E}[X \mid I = 0]\Pr[I = 0] \right)^2. \end{align*}$$