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.
I have a bachelor's degree in Japanese and a bachelor's degree in literature, and which civil service exams correspond to which major