Mathematical expectation of a random variable is one of its most important characteristics in the theory of probability. This is a concept related to the probability distribution of magnitude and is its average expected value calculated according to the formula:M = ∫xdF(x), where F(x) is the distribution function of a random variable, i.e. a function whose value at x is the probability that x belongs to the set X of values of a random variable.
This formula is called the integral of Lebesgue-Stieltjes and is based on the method of splitting the field values of integrable functions on intervals. Then it considers the integral sum.
Mathematical expectation of a discrete quantity follows directly from the Lebesgue integral-Stilties:M = Σx_i*p_i in the interval i from 1 to ∞, where x_i are the values on the discrete value, p_i elements of the set of its probability at these points. While Σp_i = 1 for I from 1 to ∞.
The mathematical expectation of the integral quantities can be derived via a generating function of the sequence. It is obvious that the integer value is a particular case of the discrete and has the following probability distribution:Σp_i = 1 for I from 0 to ∞ where p_i = P (x_i) is the probability distribution.
In order to calculate the mathematical expectation, it is necessary to differentiate P with x value equal to 1:P’(1) = Σk*p_k for k from 1 to ∞.
The generating function is a power series, the convergence of which defines mathematical expectation. When the divergence of this series of mathematical expectation is equal to infinity ∞.
To simplify the calculation of the mathematical expectation taken some of its elementary properties:- the mathematical expectation of the number is the number itself (constant);- linearity: M(a*x + b*y) = a*M(x) + b*M(y); if x ≤ y and M(y) is a finite value, then the mathematical expectation of x will also be finite, and M(x) ≤ M(y); for x = y M(x) = M(y);- mathematical expectation of the product of two quantities is equal to the product of their mathematical expectations: M(x*y) = M(x)*M(y).