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Conditional Variance

Definition (Conditional variance)

The conditional variance of an RV XX given Y=yY=y is Var(X∣Y=y)=E[X2∣Y=y]−E[X∣Y=y]2\mathrm{Var}(X|Y=y)=E[X^2|Y=y]-E[X|Y=y]^2

Theorem (Conditional Variance Formula)

We find that \mboxVar(X∣Y)\mbox{Var}(X|Y) is RV and given this information we find that \mboxVar(X)=E[\mboxVar(X∣Y)]+\mboxVar(E[X∣Y])\mbox{Var}(X)=E[\mbox{Var}(X|Y)]+\mbox{Var}(E[X|Y])