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10.2. Numerical Characteristics Theorems

Lecture



In the previous   10.2.  Numerical Characteristics Theorems We have given a number of formulas that allow us to find the numerical characteristics of functions when the laws of the distribution of arguments are known. However, in many cases, to find the numerical characteristics of functions, it is not even necessary to know the distribution laws of the arguments, it is sufficient to know only some of their numerical characteristics; at the same time, we generally do without any laws of distribution. Determining the numerical characteristics of functions by given numerical characteristics of the arguments is widely used in probability theory and can significantly simplify the solution of a number of problems. For the most part, such simplified methods relate to linear functions; however, some elementary nonlinear functions also allow a similar approach.

Present   10.2.  Numerical Characteristics Theorems we present a series of theorems on the numerical characteristics of functions, which in their totality represent a very simple apparatus for calculating these characteristics, applicable in a wide range of conditions.

1. The mathematical expectation of non-random values

 

If a   10.2.  Numerical Characteristics Theorems - non-random value, then

  10.2.  Numerical Characteristics Theorems .

The formulated property is fairly obvious; you can prove it by considering a non-random value   10.2.  Numerical Characteristics Theorems as a particular type of random, with one possible value with a probability of one; then according to the general formula for the expectation:

  10.2.  Numerical Characteristics Theorems .

2. Non-random variance

 

If a   10.2.  Numerical Characteristics Theorems - non-random value, then

  10.2.  Numerical Characteristics Theorems .

Evidence. By definition dispersion

  10.2.  Numerical Characteristics Theorems .

3. Making the non-random value a sign of the expectation

 

If a   10.2.  Numerical Characteristics Theorems - non-random value, and   10.2.  Numerical Characteristics Theorems - random, then

  10.2.  Numerical Characteristics Theorems , (10.2.1)

that is, a nonrandom value can be taken beyond the sign of the expectation.

Evidence.

a) For discontinuous quantities

  10.2.  Numerical Characteristics Theorems .

b) For continuous values

  10.2.  Numerical Characteristics Theorems .

4. The introduction of non-random values ​​for the sign of the variance and standard deviation

 

If a   10.2.  Numerical Characteristics Theorems - non-random value, and   10.2.  Numerical Characteristics Theorems - random, then

  10.2.  Numerical Characteristics Theorems , (10.2.2)

that is, it is possible to take a nonrandom value by the sign of the dispersion, squaring it.

Evidence. By definition dispersion

  10.2.  Numerical Characteristics Theorems .

The investigation

  10.2.  Numerical Characteristics Theorems ,

that is, a nonrandom value can be taken beyond the sign of the standard deviation of its absolute value. The proof will be obtained by extracting the square root from formula (10.2.2) and taking into account that sk. - substantially positive value.

5. The mathematical expectation of the sum of random variables

 

Let us prove that for any two random variables   10.2.  Numerical Characteristics Theorems and   10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems , (10.2.3)

that is, the expectation of the sum of two random variables is equal to the sum of their expectation values.

This property is known as the theorem of addition of mathematical expectations.

Evidence.

a) Let   10.2.  Numerical Characteristics Theorems - system of discontinuous random variables. Apply to the sum of random variables the general formula (10.1.6) for the mathematical expectation of a function of two arguments:

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems .

Ho   10.2.  Numerical Characteristics Theorems is nothing but the total probability that the magnitude   10.2.  Numerical Characteristics Theorems will take value   10.2.  Numerical Characteristics Theorems :

  10.2.  Numerical Characteristics Theorems ;

Consequently,

  10.2.  Numerical Characteristics Theorems .

Similarly, we prove that

  10.2.  Numerical Characteristics Theorems ,

and the theorem is proved.

b) Let   10.2.  Numerical Characteristics Theorems - system of continuous random variables. According to the formula (10.1.7)

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems . (10.2.4)

Convert the first of the integrals (10.2.4):

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems ;

similarly

  10.2.  Numerical Characteristics Theorems ,

and the theorem is proved.

It should be specially noted that the theorem of addition of mathematical expectations is valid for any random variables, both dependent and independent.

The theorem of addition of mathematical expectations is generalized to an arbitrary number of terms:

  10.2.  Numerical Characteristics Theorems , (10.2.5)

that is, the expectation of the sum of several random variables is equal to the sum of their expectation values.

For the proof, it suffices to apply the method of complete induction.

6. Mathematical expectation of a linear function

 

Consider a linear function of several random arguments.   10.2.  Numerical Characteristics Theorems :

  10.2.  Numerical Characteristics Theorems ,

Where   10.2.  Numerical Characteristics Theorems - non-random coefficients. Prove that

  10.2.  Numerical Characteristics Theorems , (10.2.6)

that is, the expectation of a linear function is equal to the same linear function of the expectation of the arguments.

Evidence. Using the addition theorem and the rule of making a nonrandom value for the sign of M. o., we obtain:

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems .

7. Disp of these sums of random variables

 

The variance of the sum of two random variables is equal to the sum of their variances plus the doubled correlation moment:

  10.2.  Numerical Characteristics Theorems . (10.2.7)

Evidence. Denote

  10.2.  Numerical Characteristics Theorems . (10.2.8)

By the theorem of addition of mathematical expectations

  10.2.  Numerical Characteristics Theorems . (10.2.9)

Let's move from random variables   10.2.  Numerical Characteristics Theorems to the corresponding centered values   10.2.  Numerical Characteristics Theorems . Subtracting term by term from equality (10.2.8) equality (10.2.9), we have:

  10.2.  Numerical Characteristics Theorems .

By definition dispersion

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems ,

Q.E.D.

Formula (10.2.7) for the variance of the sum can be generalized to any number of terms:

  10.2.  Numerical Characteristics Theorems , (10.2.10)

Where   10.2.  Numerical Characteristics Theorems - correlation moment of magnitudes   10.2.  Numerical Characteristics Theorems ,   10.2.  Numerical Characteristics Theorems sign   10.2.  Numerical Characteristics Theorems a sum means that the summation applies to all possible pairwise combinations of random variables   10.2.  Numerical Characteristics Theorems .

The proof is similar to the previous one and follows from the formula for the square of a polynomial.

Formula (10.2.10) can be written in another form:

  10.2.  Numerical Characteristics Theorems , (10.2.11)

where the double sum applies to all elements of the correlation matrix of the system of values   10.2.  Numerical Characteristics Theorems containing both correlation moments and variances.

If all random variables   10.2.  Numerical Characteristics Theorems included in the system are uncorrelated (i.e.   10.2.  Numerical Characteristics Theorems at   10.2.  Numerical Characteristics Theorems ), the formula (10.2.10) takes the form:

  10.2.  Numerical Characteristics Theorems (10.2.12)

that is, the variance of the sum of uncorrelated random variables is equal to the sum of the variances of the terms.

This position is known as the theorem of addition of variances.

8. Dispersion of a linear function

 

Consider a linear function of several random variables.

  10.2.  Numerical Characteristics Theorems ,

Where   10.2.  Numerical Characteristics Theorems - non-random values.

We prove that the variance of this linear function is expressed by the formula

  10.2.  Numerical Characteristics Theorems , (10.2.13)

Where   10.2.  Numerical Characteristics Theorems - correlation moment of magnitudes   10.2.  Numerical Characteristics Theorems ,   10.2.  Numerical Characteristics Theorems .

Evidence. We introduce the notation:

  10.2.  Numerical Characteristics Theorems .

Then

  10.2.  Numerical Characteristics Theorems . (10.2.14)

Applying to the right side of the expression (10.2.14) the formula (10.2.10) for the variance of the sum and taking into account that   10.2.  Numerical Characteristics Theorems , we get:

  10.2.  Numerical Characteristics Theorems , (10.2.15)

Where   10.2.  Numerical Characteristics Theorems - correlation moment of magnitudes   10.2.  Numerical Characteristics Theorems :

  10.2.  Numerical Characteristics Theorems .

Calculate this moment. We have:

  10.2.  Numerical Characteristics Theorems ;

similarly

  10.2.  Numerical Characteristics Theorems .

From here

  10.2.  Numerical Characteristics Theorems .

Substituting this expression in (10.2.15), we arrive at the formula (10.2.13).

In the particular case when all values   10.2.  Numerical Characteristics Theorems uncorrelated, the formula (10.2.13) takes the form:

  10.2.  Numerical Characteristics Theorems , (10.2.16)

that is, the variance of a linear function of uncorrelated random variables is equal to the sum of the products of the squares of the coefficients on the variance of the corresponding arguments.

9. The mathematical expectation of the product of random variables.

 

The expectation of the product of two random variables is equal to the product of their expectation plus the correlation moment:

  10.2.  Numerical Characteristics Theorems . (10.2.17)

Evidence. We will proceed from the definition of the correlation moment:

  10.2.  Numerical Characteristics Theorems ,

Where

  10.2.  Numerical Characteristics Theorems ;   10.2.  Numerical Characteristics Theorems .

Let us transform this expression using the properties of the expectation:

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems ,

which is obviously equivalent to the formula (10.2.17).

If random values   10.2.  Numerical Characteristics Theorems uncorrelated   10.2.  Numerical Characteristics Theorems , the formula (10.2.17) takes the form:

  10.2.  Numerical Characteristics Theorems , (10.2.18)

that is, the mathematical expectation of the product of two uncorrelated random variables is equal to the product of their mathematical expectation.

This position is known as the theorem of multiplying mathematical expectations.

Formula (10.2.17) is nothing but the expression of the second mixed central moment of the system in terms of the second mixed initial moment and mathematical expectations:

  10.2.  Numerical Characteristics Theorems . (10.2.19)

This expression is often used in practice when calculating the correlation moment in the same way as for one random variable the variance is often calculated through the second initial moment and the expectation.

The theorem of multiplying mathematical expectations is generalized to an arbitrary number of factors, only in this case it is not enough for its application that the values ​​be uncorrelated, but it is required that some higher mixed moments, the number of which depends on the number of terms in the work, vanish. These conditions are certainly fulfilled with the independence of the random variables in the product. In this case

  10.2.  Numerical Characteristics Theorems (10.2.20)

that is, the expectation of the product of independent random variables is equal to the product of their expectation.

This position is easily proved by the method of complete induction.

10. Dispersion of a product of independent random variables.

 

Let us prove that for independent quantities   10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems . (10.2.21)

Evidence. Denote   10.2.  Numerical Characteristics Theorems . By definition dispersion

  10.2.  Numerical Characteristics Theorems .

As magnitudes   10.2.  Numerical Characteristics Theorems independent   10.2.  Numerical Characteristics Theorems and

  10.2.  Numerical Characteristics Theorems .

With independent   10.2.  Numerical Characteristics Theorems magnitudes   10.2.  Numerical Characteristics Theorems also independent; Consequently,

  10.2.  Numerical Characteristics Theorems ,   10.2.  Numerical Characteristics Theorems

and

  10.2.  Numerical Characteristics Theorems . (10.2.22)

But   10.2.  Numerical Characteristics Theorems there is nothing like the second initial moment of magnitude   10.2.  Numerical Characteristics Theorems , and, therefore, is expressed through the variance:

  10.2.  Numerical Characteristics Theorems ;

similarly

  10.2.  Numerical Characteristics Theorems .

Substituting these expressions into formula (10.2.22) and citing similar terms, we arrive at formula (10.2.21).

In the case when centered random variables are multiplied (values ​​with mathematical expectation equal to zero), the formula (10.2.21) takes the form:

  10.2.  Numerical Characteristics Theorems (10.2.23)

that is, the variance of the product of independent centered random variables is equal to the product of their variances.

11. Highest moments of the sum of random variables.

 

In some cases, it is necessary to calculate the highest moments of the sum of independent random variables. Let us prove some related relations.

1) If quantities   10.2.  Numerical Characteristics Theorems independent then

  10.2.  Numerical Characteristics Theorems . (10.2.24)

Evidence.

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems ,

whence according to the theorem of multiplication of mathematical expectations

  10.2.  Numerical Characteristics Theorems .

But the first central point   10.2.  Numerical Characteristics Theorems for any value is zero; two middle terms vanish, and the formula (10.2.24) is proved.

The ratio (10.2.24) by the induction method is easily generalized to an arbitrary number of independent terms:

  10.2.  Numerical Characteristics Theorems . (10.2.25)

2) The fourth central moment of the sum of two independent random variables is expressed by the formula

  10.2.  Numerical Characteristics Theorems . (10.2.26)

Where   10.2.  Numerical Characteristics Theorems - dispersion of values   10.2.  Numerical Characteristics Theorems and   10.2.  Numerical Characteristics Theorems .

The proof is completely analogous to the previous one.

Using the complete induction method, it is easy to prove a generalization of formula (10.2.26) to an arbitrary number of independent terms:

  10.2.  Numerical Characteristics Theorems . (10.2.27)

Similar relations, if necessary, can be easily derived for moments of higher orders.

12. Addition of uncorrelated random vectors.

 

Consider on the plane   10.2.  Numerical Characteristics Theorems two uncorrelated random vectors: vector   10.2.  Numerical Characteristics Theorems with components   10.2.  Numerical Characteristics Theorems and vector   10.2.  Numerical Characteristics Theorems with components   10.2.  Numerical Characteristics Theorems (fig. 10.2.1).

  10.2.  Numerical Characteristics Theorems

Fig. 10.2.1

Consider their vector sum:

  10.2.  Numerical Characteristics Theorems ,

i.e. a vector with components:

  10.2.  Numerical Characteristics Theorems ,

  10.2.  Numerical Characteristics Theorems .

Required to determine the numerical characteristics of a random vector   10.2.  Numerical Characteristics Theorems - mathematical expectations   10.2.  Numerical Characteristics Theorems , variance and correlation moment of the components:   10.2.  Numerical Characteristics Theorems .

By the theorem of addition of mathematical expectations:

  10.2.  Numerical Characteristics Theorems ;

  10.2.  Numerical Characteristics Theorems .

According to the theorem of addition of variances

  10.2.  Numerical Characteristics Theorems ;

  10.2.  Numerical Characteristics Theorems .

Let us prove that the correlation moments also add up:

  10.2.  Numerical Characteristics Theorems , (10.2.28)

Where   10.2.  Numerical Characteristics Theorems - correlation moments of the components of each of the vectors   10.2.  Numerical Characteristics Theorems and   10.2.  Numerical Characteristics Theorems .

Evidence. By definition, the correlation moment:

  10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems . (10.2.29)

Since vectors are   10.2.  Numerical Characteristics Theorems and   10.2.  Numerical Characteristics Theorems некоррелированны, то два средних члена в формуле (10.2.29) равны нулю; два оставшихся члена представляют собой   10.2.  Numerical Characteristics Theorems and   10.2.  Numerical Characteristics Theorems ; формула (10.2.28) доказана.

Формулу (10.2.28) иногда называют «теоремой сложения корреляционных моментов».

Теорема легко обобщается на произвольное число слагаемых. Если имеется две некоррелированные системы случайных величин, т. е. два   10.2.  Numerical Characteristics Theorems -мерных случайных вектора:

  10.2.  Numerical Characteristics Theorems with components   10.2.  Numerical Characteristics Theorems ,

  10.2.  Numerical Characteristics Theorems with components   10.2.  Numerical Characteristics Theorems ,

то их векторная сумма

  10.2.  Numerical Characteristics Theorems

имеет корреляционную матрицу, элементы которой получаются суммированием элементов корреляционных матриц слагаемых:

  10.2.  Numerical Characteristics Theorems , (10.2.30)

Where   10.2.  Numerical Characteristics Theorems обозначают соответственно корреляционные моменты величин   10.2.  Numerical Characteristics Theorems ;   10.2.  Numerical Characteristics Theorems ;   10.2.  Numerical Characteristics Theorems .

Формула (10.2.30) справедлива как при   10.2.  Numerical Characteristics Theorems , так и при   10.2.  Numerical Characteristics Theorems . Действительно, составляющие вектора   10.2.  Numerical Characteristics Theorems are equal:

  10.2.  Numerical Characteristics Theorems

По теореме сложения дисперсий

  10.2.  Numerical Characteristics Theorems ,

или в других обозначениях

  10.2.  Numerical Characteristics Theorems .

По теореме сложения корреляционных моментов при   10.2.  Numerical Characteristics Theorems

  10.2.  Numerical Characteristics Theorems .

В математике суммой двух матриц называется матрица, элементы которой получены сложением соответствующих элементов этих матриц. Пользуясь этой терминологией, можно сказать, что корреляционная матрица суммы двух некоррелированных случайных векторов равна сумме корреляционных матриц слагаемых:

  10.2.  Numerical Characteristics Theorems . (10.2.31)

Это правило по аналогии с предыдущими можно назвать «теоремой сложения корреляционных матриц».


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis