Expectation value

The expectation value or mean of a random variable X or a function H(X) is given by

for a discrete or continuous variable, respectively. The sum for discrete variables is extended over all possible values of x_i, where P(X=x_i) are the corresponding probabilities. For continuous variables, the probability density is f(x). The concept is readily generalized for several random variables by replacing X by .

The expectation value is a linear operator. The expectation value of a function is sometimes written E_x(H) instead of E(H(x)).