Law of total expectation

The law of total expectation states that a random variable’s expectation equals the expectation of its conditional expectation given another variable or partition. It lets an average be computed by averaging conditional averages.

Conditional expectation: A random variable’s conditional expectation is its best mean-square prediction given specified information. The law applies expectation again to this intermediate, information-dependent average.
Expected value: Expected value is the probability-weighted average of a random variable’s possible values. The law decomposes this average into averages within cases.
Mixture distribution: A mixture distribution is a probability distribution formed by selecting among component distributions with specified probabilities. Its mean is the weighted average of the component means by total expectation.
Geometric distribution: The geometric distribution models the number of independent Bernoulli trials until the first success. Conditioning on the first trial gives a compact recursive expectation calculation.
Law of total probability: The law of total probability computes an event’s probability by summing its conditional probabilities across a partition. Both laws average conditional quantities over mutually exclusive cases.
Random variable: A random variable assigns a numerical value to each outcome of a random experiment. The identity concerns the average of a random variable over its possible outcomes.
Bayesian inference: Bayesian inference updates probability distributions using observed evidence and Bayes’ theorem. Posterior averages can be evaluated by conditioning on latent variables or observed cases.
Negative binomial distribution: The negative binomial distribution models the number of failures before a fixed number of successes in independent Bernoulli trials. Its mean can be derived by decomposing the count into waiting times between successes.
Indicator function: An indicator function equals one when a specified condition holds and zero otherwise. Indicators turn event probabilities into expectations that can be decomposed by cases.
Partition of a set: A partition divides a set into disjoint, nonempty subsets whose union is the whole set. A partition supplies the exhaustive, nonoverlapping cases used to calculate expectation.

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