Dispersion and expected value are the main characteristics of a casual event at creation of probabilistic model. These sizes are connected among themselves and in total represent a basis for the statistical analysis of sample.
Instruction
1. Any random variable has a number of the numerical characteristics determining its probability and degree of a deviation from true value. These are the initial and central moments of a different order. The first initial moment is called expected value, and the central moment of the second order – dispersion.
2. Expected value of a random variable represents its average expected value. Also this characteristic is called distribution center of probabilities and found by integration on Lebesgue-Stilties's formula: m = ∫xdf(x) where f(x) is distribution function which values are probabilities of elements of a set x ∈ X.
3. Proceeding from initial determination of integral of function, expected value can be presented in the form of the integrated sum of a numerical row which members consist of couples of elements of sets of values of a random variable and its probabilities in these points. Couples are connected by multiplication operation: m = Σxi•pi, the interval of summation makes i from 1 to ∞.
4. The given formula is a consequence from Lebesgue-Stilties's integral for a case when the analyzed size X discrete. If it integer, then it is possible to calculate expected value through the making function of the sequence which is equal to the first derivative function of distribution of probabilities at x=1: m = f’ (x) = Σk•p_k at 1 ≤ k
Dispersion of a random variable is used for assessment of a mean square of its deviation from expected value, to be exact - its dispersion around distribution center. Thus, these two sizes are connected by a formula: d = (x - m)².
Having substituted in it already known representation of expected value in a look to the integrated sum, it is possible to calculate dispersion as follows: d = Σpi • (xi - m)².
5. Dispersion of a random variable is used for assessment of a mean square of its deviation from expected value, to be exact - its dispersion around distribution center. Thus, these two sizes are connected by a formula: d = (x - m)².
6. Having substituted in it already known representation of expected value in a look to the integrated sum, it is possible to calculate dispersion as follows: d = Σpi • (xi - m)².