Expected value (mathematical expectation)

The expected value is the long-run average or mean of a random variable. This article defines discrete and continuous cases, lists key properties, examples, history, applications, and important caveats.

Expected value—also called mathematical expectation or mean—is a central concept in probability theory and statistics. Intuitively, the expected value of a random variable X describes the average outcome one would observe if an experiment were repeated many times. The operator is commonly written as E[X] or sometimes as μ. $X$

Definitions

For a discrete random variable X that takes values x with probabilities P(X = x), the expected value is the weighted sum of possible values: E[X] = Σ x·P(X = x). In the continuous case, where X has a probability density f(x), the expectation is defined by the Lebesgue integral E[X] = ∫ x f(x) dx (when this integral exists). Existence requires that the integral (or sum) of the absolute value be finite: E[|X|] < ∞. The formal definition ties closely to the notion of a random variable and the measure used to describe probabilities. $E(X)$

Basic properties

Linearity: E[aX + bY] = a E[X] + b E[Y] for constants a, b and variables X, Y (provided expectations exist).
Monotonicity: If X ≤ Y almost surely, then E[X] ≤ E[Y].
Nonmultiplicativity in general: E[XY] = E[X]E[Y] only if X and Y are independent or satisfy special relationships.
Expectation of a function: For a function g, E[g(X)] = Σ g(x)P(X = x) or ∫ g(x)f(x) dx when integrable.

These properties make the expected value a fundamental tool for calculations and theoretical developments in expectation-based arguments. $X$

Examples and uses

Common pedagogical examples include the fair die: E[die] = (1+2+3+4+5+6)/6 = 3.5, and the coin with payoff +1 heads, −1 tails: E = 0. Expected value also underlies pricing in games of chance, insurance premiums, portfolio theory, and decision theory where choices are evaluated by expected gains or losses. In statistics, sample means estimate population expectations; in applied probability, expected values appear in queueing theory and reliability calculations. $\sum _{x}xP(X=x)$ $P(X=x)$

Historical context and relation to limits

The concept of expectation arose from early studies of games of chance and correspondence between mathematicians in the 17th century. It was formalized with modern measure-theoretic probability in the 20th century. The law of large numbers connects expectation to long-run frequencies: sample averages converge to the expected value under broad conditions. This justifies interpreting expectation as the long-run average. $X$

Caveats and extensions

Not every random variable has a finite expected value. Heavy-tailed distributions such as the Cauchy distribution lack a well-defined mean; computing E[X] is either divergent or undefined. Conditional expectation E[X | Y] generalizes the idea when additional information is present and is essential in stochastic processes and Bayesian statistics. For vectors, expectation operates componentwise. Practical use requires caution: an expected value summarizes a distribution but does not capture variability or risk by itself—variance and other measures complement it. $x$ $x$ $X$

For concise mathematical statements and further reading, see foundational texts and resources on probability and statistical theory, or sources that develop measure-theoretic expectation and conditional expectation in depth. Applied examples illustrate how the same formal definition underlies diverse fields from economics to engineering.