Law of total variance

The law of total variance decomposes a random variable’s variance into the expected variance remaining after conditioning and the variance explained by the conditional mean.

Conditional expectation: The best mean-square prediction of a random variable given specified information. The conditional mean whose variance supplies the between-condition component.
Random variable: A measurable function that assigns a numerical value to each outcome of a random experiment. The identity applies to a random variable with finite second moment.
Hierarchical model: A statistical model whose parameters or observations are organized into multiple levels. It separates variability within groups from variability among group-level means.
Law of total covariance: An identity decomposing covariance into expected conditional covariance and covariance of conditional means. It generalizes the same conditional decomposition from one variable’s variance to two variables’ covariance.
Conditional variance: The variance of a random variable after specified information is held fixed. Taking its expectation gives the residual, within-condition component.
Expectation: The probability-weighted mean of a random variable. Both conditional variance and conditional-mean variance are averaged or computed through expectations.
Random-effects model: A model in which some effects are treated as random variables drawn from a population distribution. Its variance components express the same within- and between-group split.
Bias–variance decomposition: An identity splitting mean squared prediction error into squared bias, variance, and irreducible noise. It also partitions squared error, but around a target rather than across conditioning information.
Variance: The expected squared deviation of a random variable from its mean. The identity decomposes this overall measure of spread.
Sigma-algebra: A collection of measurable events closed under complements and countable unions. It formalizes the information on which conditional means and variances depend.

Linked from 3 pages