Expected

The expected value, or expectation, of a random variable is a measure of its center or the long-run average outcome in repeated trials. For a discrete variable X that can take values x1, x2, … with probabilities p1, p2, …, the expected value is E[X] = sum over i of x_i p_i. For a continuous variable with density f, E[X] = integral over all real numbers of x f(x) dx. The expectation can also be taken of a function of X: E[g(X)] = sum_i g(x_i) p_i or integral g(x) f(x) dx.

Key properties include linearity: E[aX + b] = a E[X] + b, and E[X + Y] = E[X] + E[Y], regardless of any dependence between X and Y. Existence matters: the expectation is defined only when the sums or integrals converge to a finite value.

Examples illustrate the concept. For a fair six-sided die, X takes values 1 through 6 with probability 1/6 each, so E[X] = (1+2+3+4+5+6)/6 = 3.5. If X is uniformly distributed on [0,1], E[X] = 1/2. If X is a Bernoulli(p) variable (success with probability p), E[X] = p.

Interpretation and limitations: the expectation is a long-run average, not necessarily a value the variable can take in a single trial. It may be non-integer or lie outside the observed range in some situations. The expectation summarizes the distribution’s center but does not capture variability or shape; the mean is often referred to as the population expectation, with the sample mean serving as its estimator. In decision theory, expected value (or expected utility) guides choices under uncertainty.