15.2. The concept of a random function as an extension of the concept of a system of random variables. Distribution law of a random function

Lecture



Consider some random function.   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function for a certain period of time (Fig. 15.2.1).

  15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function

Fig. 15.2.1.

Strictly speaking, we cannot depict a random function using a curve on a graph: we can only draw its specific implementations. However, for the sake of clarity, you can allow yourself to arbitrarily draw a random function in the drawing.   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function in the form of a curve, meaning by this curve is not a specific implementation, but the entire set of possible implementations   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . This convention we will mark by the fact that the curve, symbolically depicting a random function, we will carry out a dotted line.

Suppose that the course of a change in a random function is registered with the help of some device that does not record a random function continuously, but marks its values ​​at certain intervals - at times   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function .

As stated above, with a fixed value   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function a random function turns into a regular random variable. Consequently, the recording results in this case are a system   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function random variables:

  15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . (15.2.1)

Obviously, at a sufficiently high rate of operation of the recording equipment, recording a random function at such intervals will give a fairly accurate idea of ​​the course of its change. Thus, the consideration of a random function can be replaced with a certain approximation by the consideration of a system of random variables (15.2.1). As you increase   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function such a replacement is becoming more and more accurate. In the limit, the number of values ​​of the argument — and accordingly the number of random variables (15.2.1) — becomes infinite. Thus, the concept of a random function can be viewed as a natural generalization of the concept of a system of random variables to the case of an infinite (uncountable) set of quantities included in the system.

Proceeding from this interpretation of a random function, let us try to answer the question: what should be the law of distribution of a random function?

We know that the law of distribution of one random variable is a function of one argument, the law of distribution of a system of two variables is a function of two arguments, etc. However, the practical use of probabilities of functions of many arguments is so inconvenient that even for systems of three or four variables we we usually refuse to use the laws of distribution and consider only numerical characteristics. As for the law of distribution of a random function, which provides a function of an innumerable set of arguments, then such a law can at best be written in a purely formal form in some symbolic form; the practical use of such a characteristic is obviously completely excluded.

It is possible, however, for a random function to construct some probability characteristics similar to the laws of distribution. The idea of ​​building these characteristics is as follows.

Consider a random variable   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function - cross section of a random function at the moment   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function (fig. 15.2.2).

  15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function

Fig. 15.2.2.

This random variable obviously has a distribution law, which generally depends on   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . Denote it   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . Function   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function called the one-dimensional distribution law of a random function   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function .

Obviously function   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function not a complete, exhaustive characteristic of a random function   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . Indeed, this function characterizes only the distribution law   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function for this albeit arbitrary   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function ; she does not answer the question about the dependence of random variables   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function at various   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . From this point of view, a more complete characteristic of the random function   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function is the so-called two-dimensional distribution law:

  15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . (15.2.2)

This is the distribution law of a system of two random variables.   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function i.e. two arbitrary sections of a random function   15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . However, this characteristic is not exhaustive in the general case; An even more complete characteristic would be the three-dimensional law:

  15.2.  The concept of a random function as an extension of the concept of a system of random variables.  Distribution law of a random function . (15.2.3)

Obviously, it is theoretically possible to increase the number of arguments indefinitely and to obtain at the same time more and more detailed, more and more comprehensive characteristics of a random function, but it is extremely inconvenient to operate with such cumbersome characteristics depending on many arguments. Therefore, when studying the laws of distribution of random functions, it is usually limited to considering special cases where, for complete characterization of a random function, it is enough, for example, to know the function (15.2.2) (the so-called “non-response processes”).

Within the limits of the present elementary exposition of the theory of random functions, we will not use the distribution laws at all, but restrict ourselves to the consideration of the simplest characteristics of random functions similar to the numerical characteristics of random variables.


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis