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15.4. Characterization of a random function from experience

Lecture



Let over a random function   15.4.  Characterization of a random function from experience produced   15.4.  Characterization of a random function from experience independent experiments (observations) and the result was   15.4.  Characterization of a random function from experience realizations of a random function (Fig. 15.4.1).

  15.4.  Characterization of a random function from experience

Fig. 15.4.1.

Required to find estimates for the characteristics of a random function: its expectation   15.4.  Characterization of a random function from experience dispersions   15.4.  Characterization of a random function from experience and correlation function   15.4.  Characterization of a random function from experience .

To do this, consider a series of sections of a random function for the moments of time.

  15.4.  Characterization of a random function from experience

and register the values ​​accepted by the function   15.4.  Characterization of a random function from experience at these times. To each of the moments   15.4.  Characterization of a random function from experience will fit   15.4.  Characterization of a random function from experience values ​​of a random function.

Meanings   15.4.  Characterization of a random function from experience usually given equidistant; the value of the interval between adjacent values ​​is selected depending on the type of experimental curves so that the selected course can be restored from selected points. It often happens that the interval between adjacent values   15.4.  Characterization of a random function from experience it is set independently of the processing tasks of the frequency of the recording device (for example, the speed of a camera).

Registered Values   15.4.  Characterization of a random function from experience are entered into a table, each row of which corresponds to a specific implementation, and the number of columns is equal to the number of reference values ​​of the argument (Table 15.4.1).

Table 15.4.1

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

  15.4.  Characterization of a random function from experience

In table 15.4.1 in   15.4.  Characterization of a random function from experience line contains the values ​​of the random function observed in   15.4.  Characterization of a random function from experience implementation (   15.4.  Characterization of a random function from experience m experience) with argument values   15.4.  Characterization of a random function from experience . Symbol   15.4.  Characterization of a random function from experience indicated value corresponding to   15.4.  Characterization of a random function from experience implementation at the moment   15.4.  Characterization of a random function from experience .

The resulting material is nothing but results.   15.4.  Characterization of a random function from experience experiments on the system   15.4.  Characterization of a random function from experience random variables

  15.4.  Characterization of a random function from experience ,

and is processed quite similarly (see   15.4.  Characterization of a random function from experience 14.3). First of all, there are estimates for the mathematical expectation of the formula

  15.4.  Characterization of a random function from experience , (15.4.1)

then for dispersions

  15.4.  Characterization of a random function from experience (15.4.2)

and finally for the correlation moments

  15.4.  Characterization of a random function from experience . (15.4.3)

In some cases, it is convenient when calculating estimates for variances and correlation moments to use the connection between the initial and central moments and calculate them using the formulas:

  15.4.  Characterization of a random function from experience ; (15.4.4)

  15.4.  Characterization of a random function from experience . (15.4.5)

When using the latest versions of the formulas, in order to avoid the difference of close numbers, it is recommended to move the origin in advance on the ordinate axis closer to the mathematical expectation.

After these characteristics are calculated, you can, using a number of values   15.4.  Characterization of a random function from experience build dependency   15.4.  Characterization of a random function from experience (fig. 15.4.1). Similarly built dependency   15.4.  Characterization of a random function from experience . The function of two arguments   15.4.  Characterization of a random function from experience reproduced by its values ​​in a rectangular grid of dots. If necessary, all these functions are approximated by any analytical expressions.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis