Conditional variance

In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. The conditional variance of a random variable Y given the value of a random variable X is

$\operatorname {Var} (Y|X)=E((Y-E(Y|X))^{2}|X),$

where $E$ is the expectation operator. Conditional variances are important parts of ARCH models.

The law of total variance says

$\operatorname {Var} (Y)=E(Var(Y|X))+Var(E(Y|X)).$