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15.3. Characteristics of random functions

Lecture



We have had many cases to see how important the basic numerical characteristics of random variables are in probability theory: mathematical expectation and variance for one random variable, mathematical expectation and the correlation matrix for a system of random variables. The art of using numerical characteristics, leaving as far as possible the laws of distribution, is the basis of applied probability theory. The apparatus of numerical characteristics is a very flexible and powerful apparatus, which makes it relatively easy to solve many practical problems.

A completely similar apparatus is used in the theory of random functions. For random functions, the simplest basic characteristics similar to the numerical characteristics of random variables are also introduced, and the rules of actions with these characteristics are established. Such a device is sufficient to solve many practical problems.

In contrast to the numerical characteristics of random variables that provide certain numbers, the characteristics of random functions are, in general, not numbers, but functions.

Expectation of random function 15.3.  Characteristics of random functions is defined as follows. Consider the cross section of a random function 15.3.  Characteristics of random functions with fixed 15.3.  Characteristics of random functions . In this section we have the usual random variable; we define its expectation. Obviously, in the general case it depends on 15.3.  Characteristics of random functions i.e. it represents some function 15.3.  Characteristics of random functions :

15.3.  Characteristics of random functions . (15.3.1)

Thus, the mathematical expectation of a random function 15.3.  Characteristics of random functions called non-random function 15.3.  Characteristics of random functions which for every argument value 15.3.  Characteristics of random functions equal to the mathematical expectation of the corresponding cross section of a random function.

In terms of meaning, the expectation of a random function is some average function, around which the concrete implementations of the random function vary in various ways.

In fig. 15.3.1 thin lines show the implementation of a random function, a bold line is its expectation.

15.3.  Characteristics of random functions

Fig. 15.3.1.

Similarly, the variance of a random function is determined.

Variance random function 15.3.  Characteristics of random functions called non-random function 15.3.  Characteristics of random functions whose value for each 15.3.  Characteristics of random functions equal to the variance of the corresponding cross section of the random function:

15.3.  Characteristics of random functions . (15.3.2)

Variance of a random function at each 15.3.  Characteristics of random functions characterizes the scatter of possible implementations of a random function relative to the mean, in other words, the “degree of randomness” of a random function.

Obviously 15.3.  Characteristics of random functions there is a non-negative function. Taking from it the square root, we obtain the function 15.3.  Characteristics of random functions - standard deviation of the random function:

15.3.  Characteristics of random functions . (15.3.3)

Expectation and variance are very important characteristics of a random function; however, these characteristics are not enough to describe the main features of the random function. To see this, consider two random functions. 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions , visually depicted by the family of realizations in fig. 15.3.2 and 15.3.3.

15.3.  Characteristics of random functions

Fig. 15.3.2.

15.3.  Characteristics of random functions

Fig. 15.3.3.

Do random functions 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions approximately the same expectation and variance; however, the nature of these random functions is dramatically different. For random function 15.3.  Characteristics of random functions (fig. 15.3.2) is characterized by a smooth, gradual change. If, for example, at the point 15.3.  Characteristics of random functions random function 15.3.  Characteristics of random functions took a value significantly higher than the average, it is very likely that at the point 15.3.  Characteristics of random functions it will also take a value greater than average. For random function 15.3.  Characteristics of random functions characterized by a pronounced dependence between its values ​​at different 15.3.  Characteristics of random functions . In contrast, a random function 15.3.  Characteristics of random functions (fig. 15.3.3) has a sharply oscillating character with irregular, erratic fluctuations. Such a random function is characterized by a rapid decay of the dependence between its values ​​with increasing distance along 15.3.  Characteristics of random functions between them.

Obviously, the internal structure of both random processes is completely different, but this difference is not captured by either expectation or dispersion; for its description it is necessary to maintain a special characteristic. This characteristic is called the correlation function (otherwise, the autocorrelation function). The correlation function characterizes the degree of dependence between the cross sections of a random function relating to different 15.3.  Characteristics of random functions .

Let there be a random function 15.3.  Characteristics of random functions (fig. 15.3.4); Consider two of its sections relating to different points: 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions i.e. two random variables 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions . Obviously, with close values 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions magnitudes 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions are closely related: if the value 15.3.  Characteristics of random functions took some value, then the value 15.3.  Characteristics of random functions is likely to take a value close to it. It is also obvious that with an increase in the interval between sections 15.3.  Characteristics of random functions , 15.3.  Characteristics of random functions magnitude dependence 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions should generally decrease.

15.3.  Characteristics of random functions

Fig. 15.3.4.

Degree of dependence 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions can be largely characterized by their correlation point; obviously, it is a function of two arguments 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions . This function is called the correlation function.

Thus, the correlation function of the random function 15.3.  Characteristics of random functions called the non-random function of two arguments 15.3.  Characteristics of random functions which with every pair of values 15.3.  Characteristics of random functions , 15.3.  Characteristics of random functions equal to the correlation moment of the corresponding sections of the random function:

15.3.  Characteristics of random functions , (15.3.4)

Where

15.3.  Characteristics of random functions , 15.3.  Characteristics of random functions .

Let's return to examples of random functions. 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions (fig. 15.3.2 and 15.3.3). We see now that with the same expectation and variance, random functions 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions have completely different correlation functions. Correlation function of a random function 15.3.  Characteristics of random functions decreases slowly as the gap increases 15.3.  Characteristics of random functions ; on the contrary, the correlation function of a random function 15.3.  Characteristics of random functions decreases rapidly with increasing this gap.

Find out what the correlation function turns to. 15.3.  Characteristics of random functions when her arguments match. Putting 15.3.  Characteristics of random functions , we have:

15.3.  Characteristics of random functions , (15.3.5)

i.e. 15.3.  Characteristics of random functions the correlation function is drawn to the variance of the random function.

Thus, the need for dispersion as a separate characteristic of a random function disappears: as the main characteristics of a random function, it suffices to consider its expectation and correlation function.

Since the correlation moment of two random variables 15.3.  Characteristics of random functions and 15.3.  Characteristics of random functions does not depend on the sequence in which these quantities are considered, the correlation function is symmetric with respect to its arguments, i.e. it does not change when the arguments change places:

15.3.  Characteristics of random functions . (15.3.6)

If to draw the correlation function 15.3.  Characteristics of random functions in the form of a surface, then this surface will be symmetric about the vertical plane 15.3.  Characteristics of random functions passing through the bisector of the angle 15.3.  Characteristics of random functions (fig. 15.3.5).

15.3.  Characteristics of random functions

Fig. 15.3.5.

We note that the properties of the correlation function naturally follow from the properties of the correlation matrix of a system of random variables. Indeed, we replace the approximately random function. 15.3.  Characteristics of random functions system 15.3.  Characteristics of random functions random variables 15.3.  Characteristics of random functions . By increasing 15.3.  Characteristics of random functions and, accordingly, the reduction of the intervals between the arguments of the correlation matrix of the system, which is a table of two inputs, in the limit goes into the function of two continuously changing arguments, which has similar properties. The symmetry property of the correlation matrix with respect to the main diagonal is transferred to the symmetry property of the correlation function (15.3.6). On the main diagonal of the correlation matrix are the variances of random variables; similarly with 15.3.  Characteristics of random functions correlation function 15.3.  Characteristics of random functions turns into a dispersion 15.3.  Characteristics of random functions .

In practice, if you want to build a correlation function of a random function 15.3.  Characteristics of random functions , usually come in the following way: set by a number of equally spaced argument values ​​and build the correlation matrix of the resulting system of random variables. This matrix is ​​nothing but a table of values ​​of the correlation function for a rectangular grid of argument values ​​on the plane. 15.3.  Characteristics of random functions . Further, by interpolating or approximating, we can construct a function of two arguments 15.3.  Characteristics of random functions .

Instead of the correlation function 15.3.  Characteristics of random functions You can use the normalized correlation function:

15.3.  Characteristics of random functions , (15.3.7)

which is the correlation coefficient of the quantities 15.3.  Characteristics of random functions , 15.3.  Characteristics of random functions . The normalized correlation function is similar to the normalized correlation matrix of a system of random variables. With 15.3.  Characteristics of random functions normalized correlation function is equal to one:

15.3.  Characteristics of random functions . (15.3.8)

Let us find out how the basic characteristics of a random function change during elementary operations on it: when adding a nonrandom term and when multiplying by a nonrandom factor. These non-random terms and factors can be both constant values ​​and, in general, functions. 15.3.  Characteristics of random functions .

Let's add to the random function 15.3.  Characteristics of random functions nonrandom term 15.3.  Characteristics of random functions . Get a new random function:

15.3.  Characteristics of random functions . (15.3.9)

By the theorem of addition of mathematical expectations:

15.3.  Characteristics of random functions , (15.3.10)

that is, when a nonrandom term is added to a random function, the same nonrandom term is added to its expectation.

Define the correlation function of a random function 15.3.  Characteristics of random functions :

15.3.  Characteristics of random functions

15.3.  Characteristics of random functions

15.3.  Characteristics of random functions , (15.3.11)

that is, by adding a nonrandom term, the correlation function of a random function does not change.

Multiply random function 15.3.  Characteristics of random functions to a non-random factor 15.3.  Characteristics of random functions :

15.3.  Characteristics of random functions . (15.3.12)

Carrying a non-random value 15.3.  Characteristics of random functions as a sign of mathematical expectation, we have:

15.3.  Characteristics of random functions , (15.3.13)

that is, when a random function is multiplied by a nonrandom factor, its expectation is multiplied by the same factor.

Determine the correlation function:

15.3.  Characteristics of random functions

15.3.  Characteristics of random functions , (15.3.14)

that is, when multiplying a random function by a nonrandom function 15.3.  Characteristics of random functions its correlation function is multiplied by 15.3.  Characteristics of random functions .

In particular, when 15.3.  Characteristics of random functions (not dependent on 15.3.  Characteristics of random functions ), the correlation function is multiplied by 15.3.  Characteristics of random functions .

Using the derived properties of the characteristics of random functions, in some cases it is possible to significantly simplify operations with them. In particular, when it is necessary to investigate the correlation function or the variance of a random function, you can advance from it to the so-called centered function:

15.3.  Characteristics of random functions . (3/15/15)

The expectation of a centered function is identically zero, and its correlation function coincides with the correlation function of a random function 15.3.  Characteristics of random functions :

15.3.  Characteristics of random functions . (3/15/16)

In the study of issues related to the correlation properties of random functions, in the future we will always move from random functions to the corresponding centered functions, marking this with 15.3.  Characteristics of random functions at the top of the function sign.

Sometimes, besides centering, rationing of random functions is also applied. Normalized is called a random function of the form:

15.3.  Characteristics of random functions . (15.3.17)

Correlation function of the normalized random function 15.3.  Characteristics of random functions equals

15.3.  Characteristics of random functions , (15.3.18)

and its variance is one.

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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis