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14.6. Estimates for the numerical characteristics of a system of random variables

Lecture



AT   14.6.  Estimates for the numerical characteristics of a system of random variables 14.1 - 14.4 we considered the tasks associated with the estimates for the numerical characteristics of one random variable with a limited number of experiments and the construction of confidence intervals for these characteristics.

Similar questions arise when processing a limited number of observations on two or more random variables.

Here we confine ourselves to considering only point estimates for the characteristics of the system.

We first consider the case of two random variables.

There are results   14.6.  Estimates for the numerical characteristics of a system of random variables independent experiments on a random variable system   14.6.  Estimates for the numerical characteristics of a system of random variables that gave results:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ; ...;   14.6.  Estimates for the numerical characteristics of a system of random variables .

Required to find estimates for the numerical characteristics of the system: mathematical expectations   14.6.  Estimates for the numerical characteristics of a system of random variables dispersions   14.6.  Estimates for the numerical characteristics of a system of random variables and the correlation moment   14.6.  Estimates for the numerical characteristics of a system of random variables .

This question is solved in the same way as we solved it for one random variable. Unbiased estimates for mathematical expectations are arithmetic averages:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables , (14.6.1)

and for the elements of the correlation matrix -

  14.6.  Estimates for the numerical characteristics of a system of random variables (14.6.2)

The proof can be carried out similarly.   14.6.  Estimates for the numerical characteristics of a system of random variables 14.2.

When directly calculating estimates for the variances and the correlation moment, it is often convenient to use the connection between the central and initial statistical moments:

  14.6.  Estimates for the numerical characteristics of a system of random variables (14.6.3)

Where

  14.6.  Estimates for the numerical characteristics of a system of random variables (14.6.4)

Having calculated the statistical moments using formulas (14.6.3), one can then find unbiased estimates for the elements of the correlation matrix using the formulas:

  14.6.  Estimates for the numerical characteristics of a system of random variables (14.6.5)

Example. Made shots from the aircraft on the ground in single shots. The coordinates of the points of impact were recorded and the corresponding values ​​of the angle of the aircraft were recorded simultaneously. Observed values ​​of slip angle   14.6.  Estimates for the numerical characteristics of a system of random variables (in thousandths of radians) and abscissas of the point of impact   14.6.  Estimates for the numerical characteristics of a system of random variables (in meters) are given in table 14.6.1.

Table 14.6.1

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

one

-eight

-ten

eleven

+3

-one

2

+10

-2

12

-2

+4

3

+22

+4

13

+28

+12

four

+55

+10

14

+62

+20

five

+2

-one

15

-ten

-eleven

6

-39

-1+

sixteen

-eight

+2

7

-15

-eight

17

+22

+14

eight

+5

-2

18

+3

+6

9

+10

+6

nineteen

-32

-12

ten

+18

+8

20

+8

+1

Find estimates for the numerical characteristics of the system   14.6.  Estimates for the numerical characteristics of a system of random variables .

Decision. For clarity, apply all pairs of values   14.6.  Estimates for the numerical characteristics of a system of random variables on the chart (Fig. 14.6.1). The location of the points on the graph already indicates the presence of a certain dependence (positive correlation) between   14.6.  Estimates for the numerical characteristics of a system of random variables   14.6.  Estimates for the numerical characteristics of a system of random variables .

By the formulas (14.6.1) we calculate the average values ​​of   14.6.  Estimates for the numerical characteristics of a system of random variables and   14.6.  Estimates for the numerical characteristics of a system of random variables - estimates for mathematical expectations:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables .

  14.6.  Estimates for the numerical characteristics of a system of random variables

Fig. 14.6.1.

Next we find the statistical second initial moments:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables .

According to the formulas (14.6.3) we find the statistical variance:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables .

To find unbiased estimates, multiply the statistical variances by   14.6.  Estimates for the numerical characteristics of a system of random variables ; we will receive:

  14.6.  Estimates for the numerical characteristics of a system of random variables ,

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Accordingly, the standard deviations are equal to:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables .

According to the last formula (14.6.4) we find the statistical initial moment:

  14.6.  Estimates for the numerical characteristics of a system of random variables

and statistical correlation moment:

  14.6.  Estimates for the numerical characteristics of a system of random variables .

To determine the unbiased estimate, multiply it by   14.6.  Estimates for the numerical characteristics of a system of random variables ; we get:

  14.6.  Estimates for the numerical characteristics of a system of random variables ,

whence the score for the correlation coefficient is:

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Obtained relatively large value   14.6.  Estimates for the numerical characteristics of a system of random variables indicates a significant relationship between   14.6.  Estimates for the numerical characteristics of a system of random variables and   14.6.  Estimates for the numerical characteristics of a system of random variables ; on this basis, it can be assumed that the slip is the main cause of lateral deviations of the projectiles.

Let us proceed to the case of processing observations over a system of an arbitrary number of random variables.

There is a system   14.6.  Estimates for the numerical characteristics of a system of random variables random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Over system produced   14.6.  Estimates for the numerical characteristics of a system of random variables independent observations; the results of these observations are arranged in the form of a table, each row of which contains   14.6.  Estimates for the numerical characteristics of a system of random variables values ​​taken by random variables   14.6.  Estimates for the numerical characteristics of a system of random variables in one observation (table. 14.6.2).

Table 14.6.2

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

The numbers in the table and numbered by two indices are the recorded results of observations; the first index indicates the number of a random variable, the second - the number of observation, so that   14.6.  Estimates for the numerical characteristics of a system of random variables is the value accepted by   14.6.  Estimates for the numerical characteristics of a system of random variables at   14.6.  Estimates for the numerical characteristics of a system of random variables m observation.

Required to find estimates for the numerical characteristics of the system: mathematical expectations   14.6.  Estimates for the numerical characteristics of a system of random variables and elements of the correlation matrix:

  14.6.  Estimates for the numerical characteristics of a system of random variables .

On the main diagonal of the correlation matrix, obviously, there are variances of random variables.   14.6.  Estimates for the numerical characteristics of a system of random variables :

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ; ...;   14.6.  Estimates for the numerical characteristics of a system of random variables .

Estimates for mathematical expectations are found as arithmetic averages:

  14.6.  Estimates for the numerical characteristics of a system of random variables   14.6.  Estimates for the numerical characteristics of a system of random variables . (14.6.6)

Unbiased estimates for variances are determined by the formulas

  14.6.  Estimates for the numerical characteristics of a system of random variables , (14.6.7)

and for the correlation moments - according to the formulas

  14.6.  Estimates for the numerical characteristics of a system of random variables . (14.6.8)

From these data, estimates are determined for the elements of the normalized correlation matrix:

  14.6.  Estimates for the numerical characteristics of a system of random variables , (14.6.9)

Where

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables . (14.6.10)

Example. 10 batches of bombs were dropped, 5 bombs each, and hit points were recorded. The results of the experiments are summarized in table 14.6.3. In the table letter   14.6.  Estimates for the numerical characteristics of a system of random variables marked series number;   14.6.  Estimates for the numerical characteristics of a system of random variables - number of the bomb in the series.

It is required to determine the appropriate values ​​of the numerical characteristics — the expectation value and the elements of the correlation matrices — for a system of five random variables.

  14.6.  Estimates for the numerical characteristics of a system of random variables

and systems of five random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Decision. Estimates for mathematical expectations are found as arithmetic averages for the columns:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables .

When calculating the elements of the correlation matrix, we will not, as in the previous examples, use the relations between the initial and central moments; in this case, in view of the strongly varying mathematical expectations, using this technique will not give advantages. We will calculate the estimates for the moments directly using formulas (14.6.2). To do this, subtract from each element of table 14.6.3 the average value of the corresponding column. The results are summarized in table 14.6.4.

Table 14.6.3

Abscissa x

Abscissa y

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

one

2

3

four

five

one

2

3

four

five

one

-120

-20

2

60

180

-20

-15

-eight

-6

-2

2

-108

-75

-20

20

80

40

60

120

125

130

3

-200

-120

-80

-20

ten

-25

-thirty

-20

-ten

2

four

-55

-2

40

120

200

-100

-75

-35

2

2

five

five

60

100

165

220

-40

-thirty

-25

-thirty

-45

6

-240

-202

-140

-88

-thirty

80

thirty

25

ten

2

7

ten

65

120

160

205

14

25

25

thirty

ten

eight

-40

0

65

103

170

80

75

60

ten

-four

9

-100

-40

-ten

55

105

-70

-60

-thirty

-ten

0

ten

105

135

190

280

330

2

four

ten

12

four

Table 14.6.4

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables

one

2

3

four

five

one

2

3

four

five

one

-45,7

-0,1

-25,7

-25,8

33.0

-16,1

-13,4

-20,2

-19,3

-11,9

2

-33,

-55,1

-37,7

-65,8

-67,0

43.9

61.6

107.8

111.7

120.1

3

-125.7

-100,1

-107.7

-105,8

-137.0

-21,1

-28,4

-32,2

-23,3

-7.9

four

19.3

17.9

12.3

34.2

53.0

-96,1

-73,4

-47,2

-11,3

-7.9

five

79.3

79.9

72.3

79.2

73.0

-36,1

-28,4

-37,2

-43,3

-54.9

6

-165,7

-182.1

-167.7

-173,8

-177.0

83.9

31.6

12.8

-3,3

-7.9

7

84.3

84.9

92.3

74.2

58.0

17.9

26,6

12.8

16.7

0.1

eight

34.3

19.9

37.3

17.2

23.0

83.9

76.6

47.8

-3,3

-13,9

9

-25,7

-20,1

-37,7

-30,8

-42,0

-66,1

-58,4

-42,2

-23,3

-9.9

ten

179.3

154.9

162.3

194.2

183.0

5.9

5.6

-2,2

-1.3

-5.9

Squaring these numbers, summing up the columns and dividing by   14.6.  Estimates for the numerical characteristics of a system of random variables , we obtain estimates for the variances and standard deviations:

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables .

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;

  14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables ;   14.6.  Estimates for the numerical characteristics of a system of random variables .

Чтобы найти оценку для корреляционного момента, например, между величинами   14.6.  Estimates for the numerical characteristics of a system of random variables and   14.6.  Estimates for the numerical characteristics of a system of random variables составим столбец попарных произведении чисел, стоящих в первом и втором столбцах таблицы 14.6.4. Сложив все эти произведения и разделив сумму на   14.6.  Estimates for the numerical characteristics of a system of random variables , we get:

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Деля   14.6.  Estimates for the numerical characteristics of a system of random variables on   14.6.  Estimates for the numerical characteristics of a system of random variables we will receive:

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Аналогично находим все остальные элементы корреляционных матриц. Для удобства умножим все элементы обеих матриц моментов на   14.6.  Estimates for the numerical characteristics of a system of random variables . We get:

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables .

(Ввиду симметричности матриц они заполнены только наполовину.)

Нормированные корреляционные матрицы имеют вид:

  14.6.  Estimates for the numerical characteristics of a system of random variables

  14.6.  Estimates for the numerical characteristics of a system of random variables .

Рассматривая эти матрицы, убеждаемся, что величины   14.6.  Estimates for the numerical characteristics of a system of random variables находятся в весьма тесной зависимости, приближающейся к функциональной; magnitudes   14.6.  Estimates for the numerical characteristics of a system of random variables связаны менее тесно, и коэффициенты корреляции между ними убывают по мере удаления от главной диагонали корреляционной матрицы.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis