15.4. Characterization of a random function from experience

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

Fig. 15.4.1.

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

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

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

Meanings 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 it is set independently of the processing tasks of the frequency of the recording device (for example, the speed of a camera).

Registered Values 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

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

The resulting material is nothing but results. experiments on the system random variables

,

and is processed quite similarly (see 14.3). First of all, there are estimates for the mathematical expectation of the formula

, (15.4.1)

then for dispersions

(15.4.2)

and finally for the correlation moments

. (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:

; (15.4.4)

. (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 build dependency (fig. 15.4.1). Similarly built dependency . The function of two arguments reproduced by its values ​​in a rectangular grid of dots. If necessary, all these functions are approximated by any analytical expressions.