Expected value

The probability-weighted average of a random variable’s possible values. Across repeated independent trials, its sample average tends toward the expected value under suitable conditions.

Probability distribution: A mathematical description assigning probabilities to the possible outcomes of a random variable. Its probabilities and outcomes are the inputs to an expected-value calculation.
Random variable: A function that assigns a numerical value to each outcome of a random experiment. Expectation averages the values assigned by this function, weighted by their probabilities.
Expected utility hypothesis: A theory of choice in which decisions under risk maximize the probability-weighted utility of possible outcomes. It applies expectation to subjective value rather than directly to monetary outcomes.
Median: A value dividing a probability distribution or ordered dataset into two halves. Unlike the mean, the median need not shift greatly under extreme outcomes.
Discrete random variable: A random variable that takes values from a finite or countably infinite set. For discrete outcomes, expectation is a sum of value–probability products.
Probability: A numerical measure of how likely an event is, ranging from zero to one. Each possible value contributes according to its probability.
Bayes estimator: An estimator minimizing posterior expected loss under a specified prior and loss function. Its choice is determined by averaging loss over the posterior distribution.
Mode (statistics): A value or category occurring most often in data, or maximizing a distribution’s probability or density. A most likely value can differ sharply from the probability-weighted average.
Probability mass function: A function giving the probability that a discrete random variable equals each possible value. Multiplying each value by its probability and summing computes the expectation.
Weighted arithmetic mean: An average in which each value contributes in proportion to an assigned weight. Expected value is a weighted mean whose weights are outcome probabilities.

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