expectedvalue

Expected value, also called expectation or mean, is a central concept in probability and statistics that summarizes the average outcome of a random variable if the experiment could be repeated many times. For a discrete random variable X with possible values x_i and probabilities p_i, the expected value is E[X] = ∑ x_i p_i. For a continuous variable with density f, E[X] = ∫ x f(x) dx, provided the integral converges. More generally, for any function g, E[g(X)] = ∑ g(x_i) p_i or ∫ g(x) f(x) dx.

The expected value is a linear operator: E[aX + b] = a E[X] + b, and E[X + Y] = E[X] + E[Y] for any random variables X and Y, regardless of independence. If X is nonnegative, E[X] ≥ 0; the expectation may be finite, infinite, or undefined if the sum or integral diverges.

Common examples illustrate its interpretation. A fair die has E[X] = (1+2+3+4+5+6)/6 = 3.5. A fair coin that pays +1 for heads and −1 for tails has E[X] = 0. If X is an indicator of an event A, E[X] = P(A). The law of iterated expectation states that E[X] = E[ E[X|Y] ].

In practice, the expected value represents the long-run average outcome, not necessarily a typical single observation. It is a basic descriptor of a distribution and underpins many methods in statistics and decision making. The terms mean, expectation, and average are often used interchangeably, and the empirical mean estimates the expected value from data.