From AMS Glossary
(Also called expectation, mathematical expectation.) The arithmetic mean of a random variable, conceptually similar to the simple average but broader in scope.
If g(x) is a continuous function of x, then the expected value of g(x), denoted by E[g(x)], is the integral (or sum, if x is discrete) of g(x) times the probability element of x. Thus, if x is continuous with probability density function f(x) defined in the range a < x < b, then
or, if x is discrete with possible values x1, x2, . . ., xn and probability function f(xi), then
This reduces to the mean of x itself in the case g(x) = x.