Also known as expectation or mean, it is the weighted average of random variables according to probability, representing the central position of their probability distribution. Mathematical expectation is a concept that has been produced in the early development of probability theory. Most of the probability problems studied at that time were related to gambling. If someone is faced with the following situation in a gambling: among the possible outcomes such as * * * m+n, there are m outcomes that can win α, and the other n outcomes can win b), that's what he can expect in this gambling. The original form of this mathematical expectation was clearly put forward by Dutch mathematician C Huygens as early as 1657. It is a generalization of simple arithmetic average.

Let x be a discrete random variable, and the probabilities of its values x0, x 1, … are p 1, p2, … respectively, then when it is a series, its expectation is defined as. Absolute convergence of series is required here, because the expected average value should not depend on the order of summation. If x is a continuous random variable, its density function is p(x). When integrating, its expectation is defined as. In general, if X is a random variable in the probability space (ω, Fω, f, p) and its distribution function is F(x), then at that time, the expectation of defining X is the Stiglitz integral in the formula; Or the integral of the random variable x to the probability measure p on ω. However, not all random variables have expectations.

The expectation of random variables has the following properties: e (x+y) = ex+ey; If the constant α is regarded as a random variable, then eα = α; If x≥0, ex ≥ 0; If x and y are independent, then e (xy) = exey; If random variables X 1, X2, ..., Xn have joint distribution function F (X 1, X2, ..., Xn), what about a class of n-ary functions? 0? 6(x 1, x2, …, xn) (called integrable n-ary Borel measurable function, including all integrable elementary functions and continuous functions). If z = x+iy is a complex random variable, its mathematical expectation is defined as ez = ex+iey.

The above concept of mathematical expectation can also be extended to the case of random vectors. Mathematical expectations of random vectors (e.g.