SCATTERING CHARACTERISTICS

From the characteristics of the position - mathematical expectation, median, mode - let's move on to the characteristics of the spread of a random variable x. dispersion D(X)= a 2 , the standard deviation a and the coefficient of variation v. Definition and properties of dispersion for discrete random variables discussed in the previous chapter. For continuous random variables

The standard deviation is the non-negative value of the square root of the variance:

The coefficient of variation is the ratio of the standard deviation to the mathematical expectation:

Coefficient of variation - applied when M(X)> O - measures the spread in relative units, while the standard deviation - in absolute.

Example 6. For a uniformly distributed random variable X find the variance, standard deviation and coefficient of variation. The dispersion is:

Variable substitution makes it possible to write:

where With = f - aU2.

Therefore, the standard deviation is and the coefficient of variation is:

TRANSFORMATIONS OF RANDOM VALUES

For every random variable X define three more quantities - centered Y, normalized V and given U. Centered random variable Y is the difference between the given random variable X and its mathematical expectation M(X), those. Y=X - M(X). Mathematical expectation of a centered random variable Y is equal to 0, and the variance is the variance of the given random variable:

distribution function Fy(x) centered random variable Y related to the distribution function F(x) of the original random variable X ratio:

For the densities of these random variables, the equality

Normalized random variable V is the ratio of the given random variable X to its standard deviation a, i.e. V = XIo. Mathematical expectation and variance of a normalized random variable V expressed through characteristics X So:

where v is the coefficient of variation of the original random variable x. For the distribution function Fv(x) and density fv(x) normalized random variable V we have:

where F(x)- distribution function of the original random variable x; fix) is its probability density.

Reduced random variable U is a centered and normalized random variable:

For a reduced random variable

Normalized, centered and reduced random variables are constantly used both in theoretical research and in algorithms, software products, regulatory and technical and instructive and methodological documentation. In particular, because the equalities M(U) = 0, D(lf) = 1 make it possible to simplify the substantiation of methods, formulations of theorems, and calculation formulas.

Transformations of random variables and more general plan are used. So, if U = aX + b, where a and b are some numbers, then

Example 7. If a= 1/G, b = -M(X)/G, then Y is a reduced random variable, and formulas (8) are transformed into formulas (7).

With every random variable X it is possible to connect the set of random variables Y given by the formula Y = Oh + b at various a > 0 and b. This set is called scale-shear family, generated by a random variable x. Distribution functions Fy(x) constitute a scale-shift family of distributions generated by the distribution function F(x). Instead of Y= aX + b frequently used notation

Number With is called the shift parameter, and the number d- scale parameter. Formula (9) shows that X- the result of measuring a certain value - goes to K - the result of measuring the same value, if the beginning of the measurement is moved to a point With, and then use the new unit of measure, in d times greater than the old one.

For the scale-shift family (9), the distribution X called standard. In probabilistic-statistical decision-making methods and other applied research, the standard normal distribution, the standard Weibull-Gnedenko distribution, the standard gamma distribution are used.

distribution, etc. (see below).

Other transformations of random variables are also used. For example, for a positive random variable X consider Y = IgX, where IgX- decimal logarithm of a number x. Chain of equalities

relates distribution functions X and Y.

Above we got acquainted with the laws of distribution of 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 points.

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, extract from the dispersion Square root. 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, or m x .

For discrete random variables expected value :

The sum of the values of the corresponding value by the probability of random variables.

Fashion (Mod) of a random variable X is called its most probable value.

For a discrete random variable. For a continuous random variable.

Unimodal distribution

Multi modal distribution

In general, Mod and expected value not

match.

Median (Med) of a random variable X is such a value for which the probability that P(X Med). Any Med distribution can only have one.

Med divides the area under the curve into 2 equal parts. In case of unimodal and symmetrical distribution

Moments.

Most often, two types of moments are used in practice: initial and central.

Starting moment. th order of a discrete random variable X is a sum of the form:

For a continuous random variable X, the initial moment of order is the integral , it is obvious that the mathematical expectation of a random variable is the first initial moment.

Using the sign (operator) M, the initial moment of the -th order can be represented as a mat. expectation of the th power of some random variable.

Centered the random variable of the corresponding random variable X is the deviation of the random variable X from its mathematical expectation:

The mathematical expectation of a centered random variable is 0.

For discrete random variables we have:

The moments of a centered random variable are called Central moments

Central moment of order random variable X is called the mathematical expectation of the th power of the corresponding centered random variable.

For discrete random variables:

For continuous random variables:

Relation between central and initial moments of various orders

Of all the moments, the first moment (math. expectation) and the second central moment are most often used as a characteristic of a random variable.

The second central moment is called dispersion random variable. It has the designation:

By definition

For a discrete random variable:

For a continuous random variable:

The dispersion of a random variable is a characteristic of dispersion (scattering) of random variables X around its mathematical expectation.

Dispersion means scattering. The variance has the dimension of the square of a random variable.

For a visual characterization of dispersion, it is more convenient to use the value m y the same as the dimension of the random variable. For this purpose, a root is taken from the dispersion and a value is obtained, called - standard deviation (RMS) random variable X, while introducing the designation: