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

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

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

The law of total variance says

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