5.5. Numerical characteristics of random variables. Their role and purpose

In this chapter, we met a number of complete, comprehensive characteristics of random variables - the so-called distribution laws. These characteristics were:

for discrete random variable

a) distribution function;

b) the distribution series (graphically, the polygon of the distribution);

for continuous random variable:

a) distribution function;

b) distribution density (graphically - distribution curve).

Each distribution law is a function, and the indication of this function fully describes a random variable from a probabilistic point of view.

However, in many issues of practice there is no need to characterize a random variable in a complete, exhaustive manner. It is often enough to specify only certain numerical parameters, which to some extent characterize the essential features of the distribution of a random variable; any number characterizing the degree of dispersion of these values ​​relative to the average, etc. Using these characteristics, we want all the essential information about the random variable that we have to express most compactly using the minimum number of numerical parameters. Such characteristics, the purpose of which is to express in a condensed form the most essential features of the distribution, are called numerical characteristics of a random variable.

In probability theory, numerical characteristics and operations with them play a huge role. With the help of numerical characteristics it is essentially possible to solve the problem to the end, leaving aside the laws of distribution and operating with numerical characteristics alone. At the same time, the fact that when a task involves a large number of random variables, each of which has a certain effect on the numerical result of an experiment, plays a very important role, the law of distribution of this result can be considered to be largely independent of the laws of distribution of individual random variables ( called the normal distribution law). In these cases, essentially the problem for an exhaustive judgment about the resulting distribution law is not required to know the laws of the distribution of individual random variables involved in the problem; it is enough to know only some numerical characteristics of these quantities.

In probability theory and mathematical statistics, a large number of different numerical characteristics are used, having different purposes and different fields of application. Of these, in this course, we will introduce only a few, the most frequently used.