How to show that $E([X|Y])^{2} = E[X^{2}|Y]$?

Question

I have some isssue with this equation: $$ \mathrm{var}[X|Y] = E([X|Y])^{2} - (E[X|Y])^{2} $$ How to show that $ E([X|Y])^{2} = E[X^{2}|Y] $?

Answer 1

I have some isssue with this equation: $$var[X|Y] = E([X|Y])^{2} - (E[X|Y])^{2} $$

The conditional variance of $X$ measured against $Y$ is by definition:

$$\begin{align}\mathsf {Var}(X\mid Y)&=\mathsf E((X-\mathsf E(X\mid Y))^2\mid Y)\\&=\mathsf E(X^2-2X\,\mathsf E(X\mid Y)+\mathsf E(X\mid Y)^2\mid Y)\\&=\mathsf E(X^2\mid Y)-2\,\mathsf E(X\,\mathsf E(X\mid Y)\mid Y)+\mathsf E(\mathsf E(X\mid Y)^2\mid Y)\\&=\mathsf E(X^2\mid Y)-2\,\mathsf E(X\mid Y)^2+\mathsf E(X\mid Y)^2\\&=\mathsf E(X^2\mid Y)-\mathsf E(X\mid Y)^2\end{align}$$

How to show that $E([X|Y])^{2} = E[X^{2}|Y]$?

You generally do not, as it requires $\mathsf{Var}(X\mid Y)=0$, which is to say that $X$ is completely determined by $Y$ (ie there is no unexplained variance).