Expectation, Mathematical

(expected value), one of the most important characteristics of the probability distribution of a random variable. For a random variable X that assumes a sequence of values y₁, y₂, …, y_k with probabilities p₁, p₂, … p_k, respectively, mathematical expectation is defined by the formula

(assuming that the series Σ_kǀy_kǀp_k converges). Thus, for example, if X is the number of dots showing on the upper face of a die (X assumes each of the values 1, 2, 3, 4, 5, 6 with probability 1/6), then

For a random variable that has a probability density function p(y) mathematical expectation is defined by the formula

Mathematical expectation characterizes the distribution of values of a random variable. The role of mathematical expectation is fully explained by the law of large numbers. When random variables are added, their mathematical expectations are also added, and when two independent random variables are multiplied, their expectations are also multiplied. The mathematical expectation of the random variable e^ux, that is, f(t) = Ee^ux where t is a real number, is called a characteristic function.