expectation

Expectation is a probabilistic concept that quantifies the average outcome of a random variable if an experiment could be repeated many times. In everyday language, it is also used to describe what is reasonably anticipated or likely to occur.

In mathematical terms, the expectation (or expected value) of a random variable X provides a single representative value for its distribution. For a discrete X that takes values xi with probabilities pi, E[X] = sum xi pi. For a continuous X with probability density function f, E[X] = ∫ x f(x) dx, provided the integral exists (the variable is integrable and the expectation is finite).

Key properties include linearity: for any integrable random variables X and Y and real numbers a and b, E[aX + bY] = aE[X] + bE[Y]. In particular, E[X + Y] = E[X] + E[Y], and E[aX] = aE[X]. The expectation does not require independence of X and Y.

Conditional expectation, denoted E[X | Y], is a random variable that captures the expected value of X given information about Y. A fundamental result is the law of total expectation: E[X] = E[E[X | Y]].

Examples help illustrate the concept. For a fair six-sided die, E[X] = (1+2+3+4+5+6)/6 = 3.5. If a binary variable X equals 1 with probability 0.7 and 0 otherwise, E[X] = 0.7.

Related ideas include moments, variance, and, in quantum mechanics, the expectation value of an observable.