Above, we got acquainted with the laws of distribution random variables. Each distribution law exhaustively describes the properties of the probabilities of a random variable and makes it possible to calculate the probabilities of any events associated with a random variable. However, in many questions of practice there is no need for such a complete description and it is often sufficient to indicate only individual numerical parameters that characterize the essential features of the distribution. For example, the average, around which the values ​​of a random variable are scattered, is some number that characterizes the magnitude of this spread. These numbers are intended to express in a concise form the most significant features of the distribution, and are called numerical characteristics of a random variable.

Among the numerical characteristics of random variables, first of all, they consider characteristics that fix the position of a random variable on the number axis, i.e. some average value of a random variable around which its possible values ​​are grouped. Of the characteristics of the position in probability theory, the greatest role is played by expected value, which is sometimes simply called the mean value of the random variable.

Let us assume that the discrete SW?, takes the values x ( , x 2 ,..., x p with probabilities R j, p 2 ,...y Ptv those. given by the distribution series

It is possible that in these experiments the value x x observed N( times, value x 2 - N 2 times,..., value x n - N n once. At the same time + N 2 +... + N n =N.

Arithmetic mean of observation results

If a N large, i.e. N- "oh, then

describing the distribution center. The average value of a random variable obtained in this way will be called the mathematical expectation. Let us give a verbal formulation of the definition.

Definition 3.8. mathematical expectation (MO) discrete SV% is a number equal to the sum of the products of all its possible values ​​​​and the probabilities of these values ​​(notation M;):

Now consider the case when the number of possible values ​​of the discrete CV? is countable, i.e. we have RR

The formula for the mathematical expectation remains the same, only in the upper limit of the sum P is replaced by oo, i.e.

In this case, we already get a series that may diverge, i.e. the corresponding CV ^ may not have a mathematical expectation.

Example 3.8. CB?, given by the distribution series

Let's find the MO of this SW.

Solution. By definition. those. Mt, does not exist.

Thus, in the case of a countable number of SW values, we obtain the following definition.

Definition 3.9. mathematical expectation, or the average value, discrete SW, having a countable number of values, is called a number equal to the sum of a series of products of all its possible values ​​​​and the corresponding probabilities, provided that this series converges absolutely, i.e.

If this series diverges or converges conditionally, then we say that CV ^ has no mathematical expectation.

Let us pass from discrete to continuous SW with the density p(x).

Definition 3.10. mathematical expectation, or the average value, continuous SW called a number equal to

provided that this integral converges absolutely.

If this integral diverges or converges conditionally, then they say that the continuous CB? has no mathematical expectation.

Remark 3.8. If all possible values ​​of the random variable J;

belong only to the interval ( a; b) then

Mathematical expectation is not the only position characteristic used in probability theory. Sometimes such as mode and median are used.

Definition 3.11. Fashion CB ^ (designation Mot,) its most probable value is called, i.e. one for which the probability pi or probability density p(x) reaches its highest value.

Definition 3.12. Median SV?, (designation met) is called such a value for which P(t> Met) = P(? > met) = 1/2.

Geometrically, for a continuous SW, the median is the abscissa of that point on the axis Oh, for which the areas to the left and to the right of it are the same and equal to 1/2.

Example 3.9. SWt,has a distribution number

Let's find the mathematical expectation, mode and median of the SW

Solution. Mb,= 0-0.1 + 1 0.3 + 2 0.5 + 3 0.1 = 1.6. L/o? = 2. Me(?) does not exist.

Example 3.10. Continuous CB % has density

Let's find the mathematical expectation, median and mode.

Solution.

p(x) reaches a maximum, then Obviously, the median is also equal, since the areas on the right and left sides of the line passing through the point are equal.

In addition to the characteristics of the position in the theory of probability, a number of numerical characteristics for various purposes are also used. Among them, moments - initial and central - are of particular importance.

Definition 3.13. The initial moment of the kth order SW?, is called mathematical expectation k-th degree of this value: =M(t > k).

It follows from the definitions of mathematical expectation for discrete and continuous random variables that

Remark 3.9. Obviously, the initial moment of the 1st order is the mathematical expectation.

Before defining the central moment, we introduce a new concept of a centered random variable.

Definition 3.14. Centered CV is the deviation of a random variable from its mathematical expectation, i.e.

It is easy to verify that

Centering a random variable, obviously, is tantamount to transferring the origin to the point M;. The moments of a centered random variable are called central moments.

Definition 3.15. The central moment of the kth order SW % is called mathematical expectation k-th degrees of a centered random variable:

It follows from the definition of mathematical expectation that

Obviously, for any random variable ^ the central moment of the 1st order is equal to zero: with x= M(? 0) = 0.

Of particular importance for practice is the second central point from 2 . It's called dispersion.

Definition 3.16. dispersion CB?, is called the mathematical expectation of the square of the corresponding centered value (notation D?)

To calculate the variance, the following formulas can be obtained directly from the definition:

Transforming formula (3.4), we can obtain the following formula for calculating D.L.

The dispersion of SW is a characteristic scattering, the spread of values ​​of a random variable around its mathematical expectation.

The variance has the dimension of the square of a random variable, which is not always convenient. Therefore, for clarity, as a characteristic of dispersion, it is convenient to use a number whose dimension coincides with the dimension of a random variable. To do this, take the square root of the dispersion. The resulting value is called standard deviation random variable. We will denote it as a: a = l / w.

For a non-negative CB?, sometimes it is used as a characteristic the coefficient of variation, equal to the ratio of the standard deviation to the mathematical expectation:

Knowing the mathematical expectation and the standard deviation of a random variable, you can get an approximate idea of ​​the range of its possible values. In many cases, we can assume that the values ​​of the random variable % only occasionally go beyond the interval M; ± For. This rule for the normal distribution, which we will justify later, is called three sigma rule.

Mathematical expectation and variance are the most commonly used numerical characteristics of a random variable. From the definition of mathematical expectation and variance, some simple and fairly obvious properties of these numerical characteristics follow.

Protozoaproperties of mathematical expectation and dispersion.

1. Mathematical expectation of a non-random variable With equal to the value of c: M(s) = s.

Indeed, since the value With takes only one value with probability 1, then М(с) = With 1 = s.

2. The variance of the non-random variable c is equal to zero, i.e. D(c) = 0.

Really, Dc \u003d M (s - Ms) 2 \u003d M (s- c) 2 = M( 0) = 0.

3. A non-random multiplier can be taken out of the expectation sign: M(c^) = c M(?,).

Let us show the validity of this property on the example of a discrete RV.

Let RV be given by the distribution series

Then

Consequently,

The property is proved similarly for a continuous random variable.

4. A non-random multiplier can be taken out of the squared variance sign:

The more moments of a random variable are known, the more detailed idea of ​​the distribution law we have.

In probability theory and its applications, two more numerical characteristics of a random variable are used, based on the central moments of the 3rd and 4th orders - the asymmetry coefficient)