Definition

The conditional variance of an RV $X$ given $Y=y$ is $$\mbox{Var}(X|Y=y)=E[X^2|Y=y]-E[X|Y=y]^2$$ # Theorem (Conditional Variance Formula) We find that $\mbox{Var}(X|Y)$ is RV and given this information we find that $$\mbox{Var}(X)=E[\mbox{Var}(X|Y)]+\mbox{Var}(E[X|Y])$$