Conditional variance

Summary

Definition

• $X,Y$ are two random variables. Let

$$Z={\left( Y-E\left( X \right) \right)}^2$$

• We define the conditional variance of $Y$ given $X$ as

$$Var\left( X \right)=E\left( X \right)=E\left( X \right)=E\left( X \right)-{\left( E\left( X \right) \right)}^2$$

Law of total variance

$$Var\left( Y \right)=E\left( Var\left( X \right) \right)+Var\left( E\left( X \right) \right)$$