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10.3. Applications of theorems on numerical characteristics

Lecture



In this   10.3.  Applications of theorems on numerical characteristics We will demonstrate the application of the apparatus of numerical characteristics to the solution of a number of problems. Some of these tasks have independent theoretical value and will find application in the future. Other tasks are in the nature of examples and are given to illustrate the derived general formulas on a specific digital material.

Task 1. The correlation coefficient of linearly dependent random variables.

Prove that if random variables   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics linked by linear functional dependence

  10.3.  Applications of theorems on numerical characteristics ,

their correlation coefficient is   10.3.  Applications of theorems on numerical characteristics or   10.3.  Applications of theorems on numerical characteristics depending on the sign of the coefficient   10.3.  Applications of theorems on numerical characteristics .

Decision. We have:

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - variance of magnitude   10.3.  Applications of theorems on numerical characteristics .

For the correlation coefficient, we have the expression:

  10.3.  Applications of theorems on numerical characteristics . (10.3.1)

For determining   10.3.  Applications of theorems on numerical characteristics find the variance of the value   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics .

Substituting in the formula (10.3.1), we have:

  10.3.  Applications of theorems on numerical characteristics .

Magnitude   10.3.  Applications of theorems on numerical characteristics equals   10.3.  Applications of theorems on numerical characteristics when   10.3.  Applications of theorems on numerical characteristics positively and   10.3.  Applications of theorems on numerical characteristics when   10.3.  Applications of theorems on numerical characteristics negatively, as it was required to prove.

Task 2. The boundaries of the correlation coefficient.

Prove that for any random variables

  10.3.  Applications of theorems on numerical characteristics .

Decision. Consider a random variable:

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - standard deviations of magnitudes   10.3.  Applications of theorems on numerical characteristics . Determine the variance of the value   10.3.  Applications of theorems on numerical characteristics . According to the formula (10.2.13) we have:

  10.3.  Applications of theorems on numerical characteristics ,

or

  10.3.  Applications of theorems on numerical characteristics .

Since the variance of any random variable cannot be negative,

  10.3.  Applications of theorems on numerical characteristics ,

or

  10.3.  Applications of theorems on numerical characteristics ,

from where

  10.3.  Applications of theorems on numerical characteristics ,

and consequently,

  10.3.  Applications of theorems on numerical characteristics ,

Q.E.D.

Task 3. Designing a random point on a plane onto an arbitrary line.

Given a random point on a plane with coordinates   10.3.  Applications of theorems on numerical characteristics (fig. 10.3.1). Project this point on the axis   10.3.  Applications of theorems on numerical characteristics drawn through the origin at an angle   10.3.  Applications of theorems on numerical characteristics to the axis   10.3.  Applications of theorems on numerical characteristics . Point projection   10.3.  Applications of theorems on numerical characteristics on axis   10.3.  Applications of theorems on numerical characteristics there is also a random point; her distance   10.3.  Applications of theorems on numerical characteristics from the origin is a random variable. It is required to find the mean and variance   10.3.  Applications of theorems on numerical characteristics .

  10.3.  Applications of theorems on numerical characteristics

Fig.10.3.1

Decision. We have:

  10.3.  Applications of theorems on numerical characteristics .

Because   10.3.  Applications of theorems on numerical characteristics there is a linear argument function   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics then

  10.3.  Applications of theorems on numerical characteristics ;

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - variance and correlation moment values   10.3.  Applications of theorems on numerical characteristics .

Turning to the standard deviations, we get:

  10.3.  Applications of theorems on numerical characteristics . (10.3.2)

In the case of uncorrelated random variables (with   10.3.  Applications of theorems on numerical characteristics )

  10.3.  Applications of theorems on numerical characteristics . (10.3.3)

Task 4. Mathematical expectation of the number of occurrences of an event with several experiments.

Produced by   10.3.  Applications of theorems on numerical characteristics experiences, in each of which an event may or may not appear   10.3.  Applications of theorems on numerical characteristics . Probability of occurrence   10.3.  Applications of theorems on numerical characteristics at   10.3.  Applications of theorems on numerical characteristics m experience is   10.3.  Applications of theorems on numerical characteristics . Find the expected number of occurrences of the event.

Decision. Consider a discontinuous random variable   10.3.  Applications of theorems on numerical characteristics - the number of occurrences of the event in the entire series of experiments. Obviously

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - the number of occurrences of the event in the first experience

  10.3.  Applications of theorems on numerical characteristics - the number of occurrences of the event in the second experience,

……………………………………………………….

  10.3.  Applications of theorems on numerical characteristics - the number of occurrences of the event in   10.3.  Applications of theorems on numerical characteristics m experience

or shorter

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - the number of occurrences of the event in   10.3.  Applications of theorems on numerical characteristics m experience.

Each of the quantities   10.3.  Applications of theorems on numerical characteristics there is a discontinuous random variable with two possible values:   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics . Distribution range   10.3.  Applications of theorems on numerical characteristics has the form:

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

(10.3.4)

Where   10.3.  Applications of theorems on numerical characteristics - probability of non-occurrence   10.3.  Applications of theorems on numerical characteristics at   10.3.  Applications of theorems on numerical characteristics m experience.

By the theorem of addition of mathematical expectations

  10.3.  Applications of theorems on numerical characteristics , (10.3.5)

Where   10.3.  Applications of theorems on numerical characteristics - expected value   10.3.  Applications of theorems on numerical characteristics .

Calculate the expected value   10.3.  Applications of theorems on numerical characteristics . By the definition of expectation

  10.3.  Applications of theorems on numerical characteristics .

Substituting this expression into formula (10.3.5), we have

  10.3.  Applications of theorems on numerical characteristics , (10.3.6)

that is, the mathematical expectation of the number of occurrences of an event in several experiments is equal to the sum of the probabilities of the event in individual experiments.

In particular, when the conditions of the experiments are the same and

  10.3.  Applications of theorems on numerical characteristics ,

the formula (10.3.5) takes the form

  10.3.  Applications of theorems on numerical characteristics . (10.3.7)

Since the theorem of addition of mathematical expectations applies to any random variables, both dependent and independent, formulas (10.3.6) and (10.3.7) are applicable to any experiments, dependent and independent.

The derived theorem is often used in the theory of shooting, when it is required to find the average number of hits for several shots - dependent or independent. The mathematical expectation of the number of hits with several shots is equal to the sum of the probabilities of hit with individual shots.

Task 5. Dispersion of the number of occurrences of an event with several independent experiments.

Produced by   10.3.  Applications of theorems on numerical characteristics independent experiences, in each of which an event may occur   10.3.  Applications of theorems on numerical characteristics , with the probability of occurrence   10.3.  Applications of theorems on numerical characteristics at   10.3.  Applications of theorems on numerical characteristics m experience is   10.3.  Applications of theorems on numerical characteristics . Find the variance and standard deviation of the number of occurrences   10.3.  Applications of theorems on numerical characteristics .

Decision. Consider a random variable   10.3.  Applications of theorems on numerical characteristics - the number of occurrences   10.3.  Applications of theorems on numerical characteristics . Just as in the previous task, let's represent the value   10.3.  Applications of theorems on numerical characteristics as a sum:

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - the number of occurrences of the event in   10.3.  Applications of theorems on numerical characteristics m experience.

By virtue of the independence of experiments, random variables   10.3.  Applications of theorems on numerical characteristics are independent and the dispersion addition theorem is applicable to them:

  10.3.  Applications of theorems on numerical characteristics .

Find the variance of the random variable   10.3.  Applications of theorems on numerical characteristics . From the distribution series (10.3.4) we have:

  10.3.  Applications of theorems on numerical characteristics ,

from where

  10.3.  Applications of theorems on numerical characteristics , (10.3.8)

that is, the variance of the number of occurrences of an event with several independent experiments is equal to the sum of the probabilities of the occurrence and non-occurrence of an event in each experiment.

From the formula (10.3.8) we find the standard deviation of the number of occurrences of the event   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics . (10.3.9)

With unchanged experimental conditions, when   10.3.  Applications of theorems on numerical characteristics , formulas (10.3.8) and (10.3.9) are simplified and take the form:

  10.3.  Applications of theorems on numerical characteristics (10.3.10)

Task 6. Dispersion of the number of occurrences of the event in dependent experiments.

Produced by   10.3.  Applications of theorems on numerical characteristics dependent experiences in each of which an event may appear   10.3.  Applications of theorems on numerical characteristics with the probability of an event   10.3.  Applications of theorems on numerical characteristics at   10.3.  Applications of theorems on numerical characteristics m experience is   10.3.  Applications of theorems on numerical characteristics   10.3.  Applications of theorems on numerical characteristics . Determine the variance of the number of occurrences of the event.

Decision. In order to solve the problem, we again represent the number of occurrences of the event.   10.3.  Applications of theorems on numerical characteristics as a sum:

  10.3.  Applications of theorems on numerical characteristics , (10.3.11)

Where

  10.3.  Applications of theorems on numerical characteristics

Since the experiments are dependent, it is not enough for us to set the probabilities

  10.3.  Applications of theorems on numerical characteristics

that event   10.3.  Applications of theorems on numerical characteristics will occur in the first, second, third, etc., experiments. It is also necessary to specify the characteristics of the dependence of the experiments. It turns out that to solve our problem, it is enough to set the probabilities   10.3.  Applications of theorems on numerical characteristics event sharing   10.3.  Applications of theorems on numerical characteristics how in   10.3.  Applications of theorems on numerical characteristics m and so   10.3.  Applications of theorems on numerical characteristics m experience:   10.3.  Applications of theorems on numerical characteristics . Suppose these probabilities are given. Apply to expression (10.3.11) the sum dispersion theorem (formula (10.2.10)):

  10.3.  Applications of theorems on numerical characteristics , (10.3.12)

Where   10.3.  Applications of theorems on numerical characteristics - correlation moment of magnitudes   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics .

According to the formula (10.2.19)

  10.3.  Applications of theorems on numerical characteristics . (10.3.13)

Consider a random variable   10.3.  Applications of theorems on numerical characteristics . Obviously it is zero if at least one of the values   10.3.  Applications of theorems on numerical characteristics is equal to zero, i.e. at least in one of the experiments (   10.3.  Applications of theorems on numerical characteristics m or   10.3.  Applications of theorems on numerical characteristics m) event   10.3.  Applications of theorems on numerical characteristics did not appear. In order to magnitude   10.3.  Applications of theorems on numerical characteristics was equal to one, it is required that in both experiments (   10.3.  Applications of theorems on numerical characteristics m and   10.3.  Applications of theorems on numerical characteristics m) event   10.3.  Applications of theorems on numerical characteristics appeared. The probability of this is   10.3.  Applications of theorems on numerical characteristics . Consequently,

  10.3.  Applications of theorems on numerical characteristics ,

and

  10.3.  Applications of theorems on numerical characteristics .

Substituting this expression into formula (10.3.12), we get:

  10.3.  Applications of theorems on numerical characteristics . (10.3.14)

Formula (10.3.14) and expresses the variance of the number of occurrences of an event in dependent experiments. Let us analyze the structure of this formula. The first term on the right-hand side of the formula represents the variance of the number of occurrences of an event in independent experiments, and the second gives an “correction for dependence”. If the probability   10.3.  Applications of theorems on numerical characteristics equals   10.3.  Applications of theorems on numerical characteristics then this amendment is zero. If the probability   10.3.  Applications of theorems on numerical characteristics more than   10.3.  Applications of theorems on numerical characteristics This means that the conditional probability of an event   10.3.  Applications of theorems on numerical characteristics at   10.3.  Applications of theorems on numerical characteristics experience provided that   10.3.  Applications of theorems on numerical characteristics experience, it appeared more than the simple (unconditional) probability of an event occurring in   10.3.  Applications of theorems on numerical characteristics m experience   10.3.  Applications of theorems on numerical characteristics (between occurrences of an event in   10.3.  Applications of theorems on numerical characteristics m and   10.3.  Applications of theorems on numerical characteristics m experiences there is a positive correlation). If this is the case for any pair of experiments, then the correction term in formula (10.3.14) is positive and the variance of the number of occurrences of the event with dependent experiments is greater than with independent ones.

If the probability   10.3.  Applications of theorems on numerical characteristics less than   10.3.  Applications of theorems on numerical characteristics (between occurrences of an event in   10.3.  Applications of theorems on numerical characteristics m and   10.3.  Applications of theorems on numerical characteristics m experiences there is a negative correlation), then the corresponding term is negative. If this is the case for any pair of experiments, then the variance of the number of occurrences of an event with dependent experiments is less than with independent ones.

Consider a special case when   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics , ie, the conditions of all experiments are the same. The formula (10.3.14) takes the form:

  10.3.  Applications of theorems on numerical characteristics , (10.3.15)

Where   10.3.  Applications of theorems on numerical characteristics - probability of occurrence   10.3.  Applications of theorems on numerical characteristics immediately in a pair of experiments (no matter what).

In this particular case, two subcases are of particular interest:

1. The event   10.3.  Applications of theorems on numerical characteristics in any of the experiments it entails with certainty his appearance in each of the others. Then   10.3.  Applications of theorems on numerical characteristics , and the formula (10.3.15) takes the form:

  10.3.  Applications of theorems on numerical characteristics .

2. The event   10.3.  Applications of theorems on numerical characteristics in any of the experiments excludes his appearance in each of the others. Then   10.3.  Applications of theorems on numerical characteristics , и формула (10.3.15) принимает вид:

  10.3.  Applications of theorems on numerical characteristics .

Задача 7. Математическое ожидание числа объектов, приведенных в заданное состояние.

На практике часто встречается следующая задача. Имеется некоторая группа, состоящая из   10.3.  Applications of theorems on numerical characteristics объектов, по которым осуществляется какое-то воздействие. Каждый из объектов в результате воздействия может быть приведен в определенное состояние   10.3.  Applications of theorems on numerical characteristics (например, поражен, исправлен, обнаружен, обезврежен и т. п.). Probability that   10.3.  Applications of theorems on numerical characteristics -й объект будет приведен в состояние   10.3.  Applications of theorems on numerical characteristics equal to   10.3.  Applications of theorems on numerical characteristics . Найти математическое ожидание числа объектов, которые в результате воздействия по группе будут приведены в состояние   10.3.  Applications of theorems on numerical characteristics .

Decision. Свяжем с каждым из объектов случайную величину   10.3.  Applications of theorems on numerical characteristics , которая принимает значения   10.3.  Applications of theorems on numerical characteristics or   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics

Random value   10.3.  Applications of theorems on numerical characteristics - число объектов, приведенных в состояние   10.3.  Applications of theorems on numerical characteristics , - может быть представлена в виде суммы:

  10.3.  Applications of theorems on numerical characteristics .

Отсюда, пользуясь теоремой сложения математических ожиданий, получим:

  10.3.  Applications of theorems on numerical characteristics .

Математическое ожидание каждой из случайных величин   10.3.  Applications of theorems on numerical characteristics известно:

  10.3.  Applications of theorems on numerical characteristics .

Consequently,

  10.3.  Applications of theorems on numerical characteristics , (10.3.16)

т. е. математическое ожидание числа объектов, приведенных в состояние   10.3.  Applications of theorems on numerical characteristics , равно сумме вероятностей перехода в это состояние для каждого из объектов.

Особо подчеркнем, что для справедливости доказанной формулы вовсе не нужно, чтобы объекты переходили в состояние   10.3.  Applications of theorems on numerical characteristics независимо друг от друга. Формула справедлива для любого вида воздействия.

Задача 8. Дисперсия числа объектов, приведенных в заданное состояние.

Если в условиях предыдущей задачи переход каждого из объектов состояние   10.3.  Applications of theorems on numerical characteristics происходит независимо от всех других, то, применяя теорему сложения дисперсий к величине

  10.3.  Applications of theorems on numerical characteristics ,

get the variance of the number of objects given in the state   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics . (10.3.17)

If the impact on objects is made so that the transitions to the state   10.3.  Applications of theorems on numerical characteristics for individual objects are dependent, then the variance of the number of objects transferred to the state   10.3.  Applications of theorems on numerical characteristics will be expressed by the formula (see task 6)

  10.3.  Applications of theorems on numerical characteristics , (10.3.18)

Where   10.3.  Applications of theorems on numerical characteristics - the probability that as a result of exposure   10.3.  Applications of theorems on numerical characteristics th and   10.3.  Applications of theorems on numerical characteristics objects will go to state together   10.3.  Applications of theorems on numerical characteristics .

Problem 9. The mathematical expectation of the number of experiments before the   10.3.  Applications of theorems on numerical characteristics th event appears.

A number of independent experiments are carried out, in each of which an   10.3.  Applications of theorems on numerical characteristics event can occur with probability.  10.3.  Applications of theorems on numerical characteristics .Experiments are conducted until the event   10.3.  Applications of theorems on numerical characteristics appears   10.3.  Applications of theorems on numerical characteristics once, after which the experiments are terminated. Determine expected value, variance and c. ko the number of experiments   10.3.  Applications of theorems on numerical characteristics that will be made.

Decision.In Example 3   10.3.  Applications of theorems on numerical characteristics 5.7, the expectation and variance of the number of experiments before the first occurrence of an event were determined.  10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - probability of occurrence of an event in one experiment,   10.3.  Applications of theorems on numerical characteristics - probability of non- occurrence .

Consider a random variable   10.3.  Applications of theorems on numerical characteristics - the number of experiences before   10.3.  Applications of theorems on numerical characteristics the event  10.3.  Applications of theorems on numerical characteristics . It can be represented as a sum:

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - the number of experiences before the first occurrence of the event   10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics - the number of experiments from the first to the second occurrence of the event   10.3.  Applications of theorems on numerical characteristics (counting the second),

……………………………………………………………………………………………

  10.3.  Applications of theorems on numerical characteristics - the number of experiences from the   10.3.  Applications of theorems on numerical characteristics th to the   10.3.  Applications of theorems on numerical characteristics th occurrence of the event   10.3.  Applications of theorems on numerical characteristics (counting   10.3.  Applications of theorems on numerical characteristics ).

Obviously the magnitude   10.3.  Applications of theorems on numerical characteristics independent; each of them is distributed according to the same law as the first one (the number of experiments before the first occurrence of the event) and has numerical characteristics

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

Applying the theorems of addition of mathematical expectations and variances, we get:

  10.3.  Applications of theorems on numerical characteristics (10.3.19)

Task 10. The average expenditure of funds to achieve the desired result.

В предыдущей задаче был рассмотрен случай, когда предпринимается ряд опытов с целью получения вполне определенного результата -   10.3.  Applications of theorems on numerical characteristics появлений события   10.3.  Applications of theorems on numerical characteristics , которое в каждом опыте имеет одну и ту же вероятность. Эта задача является частным случаем другой, когда производится ряд опытов с целью достижения любого результата   10.3.  Applications of theorems on numerical characteristics , вероятность которого с увеличением числа опытов   10.3.  Applications of theorems on numerical characteristics возрастает по любому закону   10.3.  Applications of theorems on numerical characteristics . Предположим, что на каждый опыт расходуется определенное количество средств   10.3.  Applications of theorems on numerical characteristics . Требуется найти математическое ожидание количества средств, которое будет израсходовано.

Decision. Для того чтобы решить задачу, сначала предположим, что число производимых опытов ничем не ограничено, и что они продолжаются и после достижения результата   10.3.  Applications of theorems on numerical characteristics .Then some of these experiences will be superfluous. Let us agree to call the experience “necessary” if it is produced with a result   10.3.  Applications of theorems on numerical characteristics that has not yet been achieved , and “redundant” if it is produced with the result already achieved  10.3.  Applications of theorems on numerical characteristics .

We associate with each (   10.3.  Applications of theorems on numerical characteristics th) experience a random variable   10.3.  Applications of theorems on numerical characteristics that is zero or one, depending on whether this experience was “necessary” or “redundant”. Set

  10.3.  Applications of theorems on numerical characteristics

Consider a random variable   10.3.  Applications of theorems on numerical characteristics - the number of experiments that will have to be made to get the result   10.3.  Applications of theorems on numerical characteristics . Obviously, it can be represented as a sum:

  10.3.  Applications of theorems on numerical characteristics (10.3.20)

Of the quantities on the right-hand side of (10.3.20), the first   10.3.  Applications of theorems on numerical characteristics is nonrandom and is always equal to one (the first experience is always “necessary”). Each of the others is a random variable with possible values.  10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics . Construct a series of distribution of a random variable.   10.3.  Applications of theorems on numerical characteristics . It looks like:

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

(10.3.21)

Where   10.3.  Applications of theorems on numerical characteristics - probability of achieving the result   10.3.  Applications of theorems on numerical characteristics after   10.3.  Applications of theorems on numerical characteristics experiences.

Indeed, if the result   10.3.  Applications of theorems on numerical characteristics has already been achieved in previous   10.3.  Applications of theorems on numerical characteristics experiments, then   10.3.  Applications of theorems on numerical characteristics (experience is redundant), if not achieved, then   10.3.  Applications of theorems on numerical characteristics (experience is necessary).

Find the expected value   10.3.  Applications of theorems on numerical characteristics . From the distribution series (10.3.21) we have:

  10.3.  Applications of theorems on numerical characteristics .

It is easy to make sure that the same formula will be valid when   10.3.  Applications of theorems on numerical characteristics , because   10.3.  Applications of theorems on numerical characteristics .

Apply to expression (10.3.20) the theorem of addition of mathematical expectations. We get:

  10.3.  Applications of theorems on numerical characteristics

or denoting   10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics . (10.3.22)

Every experience requires an expense   10.3.  Applications of theorems on numerical characteristics . Multiplying the resulting value   10.3.  Applications of theorems on numerical characteristics on   10.3.  Applications of theorems on numerical characteristics , we define the average cost of funds to achieve the result   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics . (10.3.23)

This formula is derived under the assumption that the value of each experience is the same. If this is not the case, then another method can be applied - to present the total expenditure of funds   10.3.  Applications of theorems on numerical characteristics as the sum of the costs of performing individual experiments, which takes two values:  10.3.  Applications of theorems on numerical characteristics , if a   10.3.  Applications of theorems on numerical characteristics experience is “necessary,” and zero if it is “redundant.” The average expense will be   10.3.  Applications of theorems on numerical characteristics presented in the form:

  10.3.  Applications of theorems on numerical characteristics . (10.3.24)

Problem 11. The mathematical expectation of the sum of a random number of random terms.

In a number of practical applications of probability theory one has to meet with sums of random variables, in which the number of terms is unknown in advance, by chance.

Let's set the following task. Random value   10.3.  Applications of theorems on numerical characteristics represents the sum   10.3.  Applications of theorems on numerical characteristics random variables:

  10.3.  Applications of theorems on numerical characteristics , (10.3.25)

and   10.3.  Applications of theorems on numerical characteristics - also a random variable. Suppose that we know the mathematical expectations of   10.3.  Applications of theorems on numerical characteristics all terms:

  10.3.  Applications of theorems on numerical characteristics

и что величина   10.3.  Applications of theorems on numerical characteristics не зависит ни от одной из величин   10.3.  Applications of theorems on numerical characteristics .

Требуется найти математическое ожидание величины   10.3.  Applications of theorems on numerical characteristics .

Decision. Число слагаемых в сумме есть дискретная случайная величина. Предположим, что нам известен ее ряд распределения:

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

Where   10.3.  Applications of theorems on numerical characteristics - вероятность того, что величина   10.3.  Applications of theorems on numerical characteristics took meaning   10.3.  Applications of theorems on numerical characteristics . Зафиксируем значение   10.3.  Applications of theorems on numerical characteristics и найдем при этом условии математическое ожидание величины   10.3.  Applications of theorems on numerical characteristics (условное математическое ожидание):

  10.3.  Applications of theorems on numerical characteristics . (10.3.26)

Теперь применим формулу полного математического ожидания, для чего умножим каждое условное математическое ожидание на вероятность соответствующей гипотезы   10.3.  Applications of theorems on numerical characteristics и сложим:

  10.3.  Applications of theorems on numerical characteristics . (10.3.27)

Особый интерес представляет случай, когда все случайные величины   10.3.  Applications of theorems on numerical characteristics имеют одно и то же математическое ожидание:

  10.3.  Applications of theorems on numerical characteristics .

Тогда формула (10.3.26) принимает вид:

  10.3.  Applications of theorems on numerical characteristics

and

  10.3.  Applications of theorems on numerical characteristics . (10.3.28)

Сумма в выражении (10.3.28) представляет собой не что иное, как математическое ожидание величины   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics .

From here

  10.3.  Applications of theorems on numerical characteristics , (10.3.29)

т. е. математическое ожидание суммы случайного числа случайных слагаемых с одинаковыми средними значениями (если только число слагаемых не зависит от их значений) равно произведению среднего значения каждого из слагаемых на среднее число слагаемых.

Снова отметим, что полученный результат справедлив как для независимых, так и для зависимых слагаемых   10.3.  Applications of theorems on numerical characteristics лишь бы число слагаемых   10.3.  Applications of theorems on numerical characteristics не зависело от самих слагаемых.

Ниже мы решим ряд конкретных примеров из разных областей практики, на которых продемонстрируем конкретное применение общих методов оперирования с числовыми характеристиками, вытекающих из доказанных теорем, и специфических приемов, связанных с решенными выше общими задачами.

Пример 1. Монета бросается 10 раз. Определить математическое ожидание и среднее квадратическое отклонение числа   10.3.  Applications of theorems on numerical characteristics выпавших гербов.

Decision. По формулам (10.3.7) и (10.3.10) найдем:

  10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics .

Пример 2. Производится 5 независимых выстрелов по круглой мишени диаметром 20 см. Прицеливание - по центру мишени, систематическая ошибка отсутствует, рассеивание - круговое, среднее квадратическое отклонение   10.3.  Applications of theorems on numerical characteristics см. Найти математическое ожидание и с. к. о. числа попаданий.

Decision. Вероятность попадания в мишень при одном выстреле вычислим по формуле (9.4.5):

  10.3.  Applications of theorems on numerical characteristics .

Пользуясь формулами (10.3.7) и (10.3.10), получим:

  10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics .

Example 3. Airborne attack is carried out, in which 20 type 1 aircraft and 30 type 2 aircraft are involved. Type 1 aircraft are attacked by fighter aircraft. The number of attacks per device is subject to the Poisson law with the parameter  10.3.  Applications of theorems on numerical characteristics . Each attack of a fighter aircraft type 1 is affected with probability   10.3.  Applications of theorems on numerical characteristics .Type 2 aircraft are attacked by anti-aircraft missiles. The number of missiles sent to each device is subject to Poisson’s law with a parameter   10.3.  Applications of theorems on numerical characteristics ; each missile hits a type 2 aircraft with probability  10.3.  Applications of theorems on numerical characteristics . All devices that are part of the raid are attacked and attacked independently of each other. To find:

1) expectation, variance, and c. ko the number of type 1 aircraft affected;

2) expectation, variance, and c. ko the number of type 2 aircraft affected;

3) expectation, variance, and c. ko the numbers of affected aircraft of both types.

Decision. Instead of “the number of attacks” on each device of type 1, we consider “the number of attacking attacks”, also distributed according to the Poisson law, but with a different parameter:

  10.3.  Applications of theorems on numerical characteristics .

The probability of hitting each of the aircraft of type 1 will be equal to the probability that it will have at least one striking attack:

  10.3.  Applications of theorems on numerical characteristics .

The probability of damage to each of the aircraft type 2 will find the same:

  10.3.  Applications of theorems on numerical characteristics .

The mathematical expectation of the number of affected devices of type 1 will be:

  10.3.  Applications of theorems on numerical characteristics .

Dispersion and with. ko this number:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

Expectation, number variance and c. ko Type 2 affected devices:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

Mathematical expectation, variance and c. ko total number of affected devices of both types:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

Example 4. Random variables   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics represent the elementary errors that occur at the input of the device. They have mathematical expectations   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics dispersions   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics ; the correlation coefficient of these errors is   10.3.  Applications of theorems on numerical characteristics . The error at the output of the device is connected with errors at the input by the functional dependence:

  10.3.  Applications of theorems on numerical characteristics .

Find the expected value of the error at the output of the device.

Decision.

  10.3.  Applications of theorems on numerical characteristics .

Using the connection between the initial and central moments and formula (10.2.17), we have:

  10.3.  Applications of theorems on numerical characteristics ;

  10.3.  Applications of theorems on numerical characteristics ;

  10.3.  Applications of theorems on numerical characteristics ,

from where

  10.3.  Applications of theorems on numerical characteristics .

Example 5. An airplane bombing a motorway with a width of 30 m (Fig. 10.3.2). Flight direction is angle   10.3.  Applications of theorems on numerical characteristics with the direction of the motorway. Aiming - on the middle line of the motorway, systematic errors are absent. Dispersion given by the main probable deviations: in the direction of flight   10.3.  Applications of theorems on numerical characteristics m and in the lateral direction   10.3.  Applications of theorems on numerical characteristics m. Find the probability of hitting the highway when dropping a single bomb.

  10.3.  Applications of theorems on numerical characteristics

Fig. 10.3.2

Decision. Design a random point of impact on the axis   10.3.  Applications of theorems on numerical characteristics perpendicular to the motorway and apply the formula (10.3.3). It obviously remains fair if we substitute probable deviations instead of the mean-square ones:

  10.3.  Applications of theorems on numerical characteristics .

From here

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

We will find the probability of hitting the highway by the formula (6.3.10):

  10.3.  Applications of theorems on numerical characteristics .

Note. The method used here for recalculating scattering to other axes is only suitable for calculating the probability of hitting a region in the form of a band; for a rectangle whose sides are turned at an angle to the scattering axes, it is no longer suitable. The probability of hitting each of the lanes whose intersection forms a rectangle can be calculated using this technique, but the probability of hitting the rectangle is no longer equal to the product of the probabilities of hitting the lanes, since these events are dependent.

Example 6. A group of objects is monitored using a radar system for some time; the group consists of four objects; each one in time   10.3.  Applications of theorems on numerical characteristics is found with a probability equal to:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

Find the expectation of the number of objects that will be detected in time.   10.3.  Applications of theorems on numerical characteristics .

Decision. By the formula (10.3.16) we have:

  10.3.  Applications of theorems on numerical characteristics .

Example 7. A series of events are taken, each of which, if it takes place, brings random net income.   10.3.  Applications of theorems on numerical characteristics normal distributed with average   10.3.  Applications of theorems on numerical characteristics (conventional units). The number of events for a given period of time is random and distributed according to the law.

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

and does not depend on the income generated by the events. Determine the average expected income for the entire period.

Decision. Based on task 11 of this   10.3.  Applications of theorems on numerical characteristics find the expectation of total income   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - the average income from one event,   10.3.  Applications of theorems on numerical characteristics - average expected number of events. We have:

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics .

Example 8. Device error is expressed by function.

  10.3.  Applications of theorems on numerical characteristics (10.3.30)

Where   10.3.  Applications of theorems on numerical characteristics - the so-called "primary errors", representing a system of random variables (random vector).

Random vector   10.3.  Applications of theorems on numerical characteristics characterized by mathematical expectations

  10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics

and correlation matrix:

  10.3.  Applications of theorems on numerical characteristics .

Determine the mean, variance and standard deviation of the instrument error.

Decision. Since the function (10.3.30) is linear, applying formulas (10.2.6) and (10.2.13), we find:

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics .

Example 9. To detect the source of a malfunction in a computer, tests are carried out. In each sample, failure is independent of other samples is localized with probability   10.3.  Applications of theorems on numerical characteristics . On average, each sample takes 3 minutes. Find the expectation of the time it takes to isolate the problem.

Decision. Using the result of task 9 given   10.3.  Applications of theorems on numerical characteristics (mathematical increase in the number of experiments to   10.3.  Applications of theorems on numerical characteristics th event occurrence   10.3.  Applications of theorems on numerical characteristics ), believing   10.3.  Applications of theorems on numerical characteristics find the average number of samples

  10.3.  Applications of theorems on numerical characteristics .

These five samples will require an average of

  10.3.  Applications of theorems on numerical characteristics (minutes).

Example 10. Shooting at a fuel tank is performed. The probability of hitting each shot is 0.3. The shots are independent. At the first hit in the reservoir, only the fuel flow appears, at the second hit the fuel is ignited. After the ignition of the fuel, the shooting stops. Find the mathematical expectation of the number of shots fired.

Decision. Using the same formula as in the previous example, we find the mathematical expectation of the number of shots before the 2nd hit:

  10.3.  Applications of theorems on numerical characteristics .

Example 11. The probability of an object being detected by a radar with an increase in the number of review cycles increases according to the law

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - the number of cycles since the beginning of the observation.

Find the mathematical expectation of the number of cycles after which the object will be detected.

Decision. Using the results of task 10 of this section, we obtain:

  10.3.  Applications of theorems on numerical characteristics .

Example 12. In order to perform a specific task of gathering information, several scouts are sent to a given area. Each scout sent to the area of ​​destination with a probability of 0.7. To complete the task, it is sufficient to have three intelligence officers in the area. One scout cannot cope with the task at all, and two scouts perform it with a probability of 0.4. Continuous communication with the area is ensured, and additional scouts are sent only if the task has not yet been completed.

It is required to find the expectation of the number of intelligence officers to be sent.

Decision. Denote   10.3.  Applications of theorems on numerical characteristics - The number of scouts arrived in the area, which was sufficient to complete the task. In task 10 of this   10.3.  Applications of theorems on numerical characteristics the mathematical expectation of the number of experiments was found, which is needed in order to achieve a certain result, the probability of which increases with the increase in the number of experiments according to the law   10.3.  Applications of theorems on numerical characteristics . This expectation is:

  10.3.  Applications of theorems on numerical characteristics .

In our case:

  10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics ;   10.3.  Applications of theorems on numerical characteristics ;

  10.3.  Applications of theorems on numerical characteristics .

Expectation value   10.3.  Applications of theorems on numerical characteristics equally:

  10.3.  Applications of theorems on numerical characteristics .

So, in order for the task to be completed, it is necessary that an average of 2.6 scouts arrive in the area.

Now we solve the following problem. How many scouts will on average have to be sent to the area in order for them to arrive on average   10.3.  Applications of theorems on numerical characteristics ?

Let sent   10.3.  Applications of theorems on numerical characteristics scouts. The number of arrived scouts can be represented as

  10.3.  Applications of theorems on numerical characteristics ,

where is the random variable   10.3.  Applications of theorems on numerical characteristics takes the value 1 if   10.3.  Applications of theorems on numerical characteristics th scout arrived, and 0 if not arrived. Magnitude   10.3.  Applications of theorems on numerical characteristics is nothing but the sum of a random number of random terms (see Problem 11 of this   10.3.  Applications of theorems on numerical characteristics ). With this in mind, we have:

  10.3.  Applications of theorems on numerical characteristics ,

from where

  10.3.  Applications of theorems on numerical characteristics ,

but   10.3.  Applications of theorems on numerical characteristics where   10.3.  Applications of theorems on numerical characteristics - probability of arrival of the dispatched scout (in our case   10.3.  Applications of theorems on numerical characteristics ). Magnitude   10.3.  Applications of theorems on numerical characteristics we have just found and equal to 2.6 We have:

  10.3.  Applications of theorems on numerical characteristics .

Example 13. A radar station looks in the area of ​​space in which it is located   10.3.  Applications of theorems on numerical characteristics objects. In one review cycle, it detects each of the objects (independently of the other cycles) with probability   10.3.  Applications of theorems on numerical characteristics . One cycle takes time   10.3.  Applications of theorems on numerical characteristics . How long will it take to   10.3.  Applications of theorems on numerical characteristics objects detect on average   10.3.  Applications of theorems on numerical characteristics ?

Decision. We first find the expectation of the number of objects detected after   10.3.  Applications of theorems on numerical characteristics review cycles. Behind   10.3.  Applications of theorems on numerical characteristics cycles one (any) of the objects is detected with probability

  10.3.  Applications of theorems on numerical characteristics ,

and the average number of objects found in   10.3.  Applications of theorems on numerical characteristics cycles, by the expectation addition theorem (see problem 5 of this   10.3.  Applications of theorems on numerical characteristics ) is equal to:

  10.3.  Applications of theorems on numerical characteristics .

Putting

  10.3.  Applications of theorems on numerical characteristics ,

get the required number of cycles   10.3.  Applications of theorems on numerical characteristics from the equation

  10.3.  Applications of theorems on numerical characteristics ,

deciding which, we find:

  10.3.  Applications of theorems on numerical characteristics ,

where does the time required for detection on average   10.3.  Applications of theorems on numerical characteristics objects will be equal to:

  10.3.  Applications of theorems on numerical characteristics .

Example 14. Let us change the conditions of Example 13. Let the radar station monitor the area only until it is detected.   10.3.  Applications of theorems on numerical characteristics objects, after which the observation stops or continues in the new mode. Find the expectation of the time it takes.

In order to solve this problem, it is not enough to ask the probability of detecting one object in one cycle, but you also need to indicate how the probability of   10.3.  Applications of theorems on numerical characteristics objects will be detected at least   10.3.  Applications of theorems on numerical characteristics . The easiest way to calculate this probability, if we assume that objects are detected independently of each other. We make this assumption and solve the problem.

Decision. With independent detections you can monitor   10.3.  Applications of theorems on numerical characteristics objects represent how   10.3.  Applications of theorems on numerical characteristics independent experiences. After   10.3.  Applications of theorems on numerical characteristics cycles, each of the objects is detected with probability

  10.3.  Applications of theorems on numerical characteristics .

Probability that after   10.3.  Applications of theorems on numerical characteristics cycles will be detected at least   10.3.  Applications of theorems on numerical characteristics objects from   10.3.  Applications of theorems on numerical characteristics , we find by the theorem on the repetition of experiments:

  10.3.  Applications of theorems on numerical characteristics .

The average number of cycles, after which will be detected at least   10.3.  Applications of theorems on numerical characteristics objects, determined by the formula (10.3.22):

  10.3.  Applications of theorems on numerical characteristics .

Example 15. On a plane   10.3.  Applications of theorems on numerical characteristics random point   10.3.  Applications of theorems on numerical characteristics with coordination   10.3.  Applications of theorems on numerical characteristics deviates from the required position (origin) under the influence of three independent vector errors   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics . Each of the vectors is characterized by two components:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics

(fig. 10.3.3). The numerical characteristics of these three vectors are:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

  10.3.  Applications of theorems on numerical characteristics

Fig. 10.3.3

Найти характеристики суммарной ошибки (вектора, отклоняющего точку   10.3.  Applications of theorems on numerical characteristics от начала координат).

Decision. Применяя теоремы сложения математических ожиданий, дисперсий и корреляционных моментов, получим:

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

Where

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

from where

  10.3.  Applications of theorems on numerical characteristics

and

  10.3.  Applications of theorems on numerical characteristics .

Пример 16. Тело, которое имеет форму прямоугольного параллелепипеда с размерами   10.3.  Applications of theorems on numerical characteristics летит в пространстве, беспорядочно вращаясь вокруг центра массы так, что все его ориентации одинаково вероятны. Тело находится в потоке частиц, и среднее число частиц, встречающихся с телом, пропорционально средней площади, которую тело подставляет потоку. Найти математическое ожидание площади проекции тела на плоскость, перпендикулярную направлению его движения.

Decision. Так как все ориентации тела в пространстве одинаково вероятны, то направление плоскости проекций безразлично. Очевидно, площадь проекции тела равна половине суммы проекций всех граней параллелепипеда (так как каждая точка проекции представляет собой проекцию двух точек на поверхности тела). Применяя теорему сложения математических ожиданий и формулу для средней площади проекции плоской фигуры (см. пример 3   10.3.  Applications of theorems on numerical characteristics 10.1), получим:

  10.3.  Applications of theorems on numerical characteristics ,

Where   10.3.  Applications of theorems on numerical characteristics - полная площадь поверхности параллелепипеда.

Заметим, что выведенная формула справедлива не только для параллелепипеда, но и для любого выпуклого тела: средняя площадь проекции такого тела при беспорядочном вращении равна одной четверти полной его поверхности. Рекомендуем читателю в качестве упражнения доказать это положение.

Пример 17. На оси абсцисс   10.3.  Applications of theorems on numerical characteristics движется случайным образом точка   10.3.  Applications of theorems on numerical characteristics по следующему закону. В начальный момент она находится в начале координат и начинает двигаться с вероятностью   10.3.  Applications of theorems on numerical characteristics вправо и с вероятностью   10.3.  Applications of theorems on numerical characteristics to the left. Пройдя единичное расстояние, точка с вероятностью   10.3.  Applications of theorems on numerical characteristics продолжает двигаться в том же направлении, а с вероятностью   10.3.  Applications of theorems on numerical characteristics меняет его на противоположное. Пройдя единичное расстояние, точка снова с вероятностью   10.3.  Applications of theorems on numerical characteristics продолжает движение в том направлении, в котором двигалась, а с вероятностью   10.3.  Applications of theorems on numerical characteristics меняет его на противоположное и т. д.

В результате такого случайного блуждания по оси абсцисс точка   10.3.  Applications of theorems on numerical characteristics after   10.3.  Applications of theorems on numerical characteristics шагов займет случайное положение, которое мы обозначим   10.3.  Applications of theorems on numerical characteristics . Требуется найти характеристики случайной величины   10.3.  Applications of theorems on numerical characteristics : математическое ожидание и дисперсию.

Decision. Прежде всего, из соображений симметрии задачи ясно, что   10.3.  Applications of theorems on numerical characteristics . Чтобы найти   10.3.  Applications of theorems on numerical characteristics , представим   10.3.  Applications of theorems on numerical characteristics в виде суммы   10.3.  Applications of theorems on numerical characteristics слагаемых:

  10.3.  Applications of theorems on numerical characteristics , (10.3.31)

Where   10.3.  Applications of theorems on numerical characteristics - расстояние, пройденное точкой на   10.3.  Applications of theorems on numerical characteristics -м шаге, т. е.   10.3.  Applications of theorems on numerical characteristics , если точка двигалась на этом шаге вправо, и   10.3.  Applications of theorems on numerical characteristics , если она двигалась влево.

По теореме о дисперсии суммы (см. формулу (10.2.10)) имеем:

  10.3.  Applications of theorems on numerical characteristics .

It's clear that   10.3.  Applications of theorems on numerical characteristics , так как величина   10.3.  Applications of theorems on numerical characteristics takes values   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics с одинаковой вероятностью (из тех же соображений симметрии). Найдем корреляционные моменты

  10.3.  Applications of theorems on numerical characteristics .

Начнем со случая   10.3.  Applications of theorems on numerical characteristics , когда величины   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics стоят рядом в сумме (10.3.31). It's clear that   10.3.  Applications of theorems on numerical characteristics принимает значение   10.3.  Applications of theorems on numerical characteristics with probability   10.3.  Applications of theorems on numerical characteristics и значение   10.3.  Applications of theorems on numerical characteristics with probability   10.3.  Applications of theorems on numerical characteristics . We have:

  10.3.  Applications of theorems on numerical characteristics .

Рассмотрим, далее, случай   10.3.  Applications of theorems on numerical characteristics . В этом случае произведение   10.3.  Applications of theorems on numerical characteristics equally   10.3.  Applications of theorems on numerical characteristics , если оба перемещения - на   10.3.  Applications of theorems on numerical characteristics -м и   10.3.  Applications of theorems on numerical characteristics -м шаге - происходят в одном и том же направлении. Это может произойти двумя способами. Или точка   10.3.  Applications of theorems on numerical characteristics все три шага -   10.3.  Applications of theorems on numerical characteristics -й,   10.3.  Applications of theorems on numerical characteristics th and   10.3.  Applications of theorems on numerical characteristics -й - двигалась в одном и том же направлении, или же она дважды изменила за эти три шага свое направление. Найдем вероятность того, что   10.3.  Applications of theorems on numerical characteristics :

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics .

Найдем теперь вероятность того, что   10.3.  Applications of theorems on numerical characteristics . Это тоже может произойти двумя способами: или точка изменила свое направление при переводе от   10.3.  Applications of theorems on numerical characteristics -го шага к   10.3.  Applications of theorems on numerical characteristics -му, а при переходе от   10.3.  Applications of theorems on numerical characteristics -го шага к   10.3.  Applications of theorems on numerical characteristics -му сохранила его, или наоборот. We have:

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics .

Thus, the magnitude   10.3.  Applications of theorems on numerical characteristics имеет два возможных значения   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics , которые она принимает с вероятностями соответственно   10.3.  Applications of theorems on numerical characteristics and   10.3.  Applications of theorems on numerical characteristics .

Ее математическое ожидание равно:

  10.3.  Applications of theorems on numerical characteristics .

Легко доказать по индукции, что для любого расстояния   10.3.  Applications of theorems on numerical characteristics между шагами в ряду   10.3.  Applications of theorems on numerical characteristics справедливы формулы:

  10.3.  Applications of theorems on numerical characteristics ,

  10.3.  Applications of theorems on numerical characteristics ,

and therefore

  10.3.  Applications of theorems on numerical characteristics .

Таким образом, корреляционная матрица системы случайных величин   10.3.  Applications of theorems on numerical characteristics будет иметь вид:

  10.3.  Applications of theorems on numerical characteristics .

Random Variance   10.3.  Applications of theorems on numerical characteristics будет равна:

  10.3.  Applications of theorems on numerical characteristics ,

или же, производя суммирование элементов, стоящих на одном расстоянии от главной диагонали,

  10.3.  Applications of theorems on numerical characteristics .

Пример 18. Найти асимметрию биномиального распределения

  10.3.  Applications of theorems on numerical characteristics   10.3.  Applications of theorems on numerical characteristics . (10.3.32)

Decision. Известно, что биномиальное распределение (10.3.32) представляет собой распределение числа появлений в   10.3.  Applications of theorems on numerical characteristics независимых опытах некоторого события, которое в одном опыте имеет вероятность   10.3.  Applications of theorems on numerical characteristics . Представим случайную величину   10.3.  Applications of theorems on numerical characteristics - число появлений события в   10.3.  Applications of theorems on numerical characteristics опытах - как сумму   10.3.  Applications of theorems on numerical characteristics random variables:

  10.3.  Applications of theorems on numerical characteristics ,

Where

  10.3.  Applications of theorems on numerical characteristics

По теореме сложения третьих центральных моментов

  10.3.  Applications of theorems on numerical characteristics . (10.3.33)

Найдем третий центральный момент случайной величины   10.3.  Applications of theorems on numerical characteristics . Она имеет распределения

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

  10.3.  Applications of theorems on numerical characteristics

Третий центральный момент величины   10.3.  Applications of theorems on numerical characteristics equals:

  10.3.  Applications of theorems on numerical characteristics .

Подставляя в (10.3.33), получим:

  10.3.  Applications of theorems on numerical characteristics .

Чтобы получить асимметрию, нужно разделить третий центральный момент величины   10.3.  Applications of theorems on numerical characteristics на куб среднего квадратического отклонения:

  10.3.  Applications of theorems on numerical characteristics .

Пример 19. Имеется   10.3.  Applications of theorems on numerical characteristics положительных, одинаково распределенных независимых случайных величин:

  10.3.  Applications of theorems on numerical characteristics .

Найти математическое ожидание случайной величины

  10.3.  Applications of theorems on numerical characteristics .

Decision. Ясно, что математическое ожидание величины   10.3.  Applications of theorems on numerical characteristics существует, так как она заключена между нулем и единицей. Кроме того, легко видеть, что закон распределения системы величин   10.3.  Applications of theorems on numerical characteristics , каков бы он ни был, симметричен относительно своих переменных, т. е. не меняется при любой их перестановке. Рассмотрим случайные величины:

  10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics ,   10.3.  Applications of theorems on numerical characteristics .

Очевидно, их закон распределения тоже должен обладать свойством симметрии, т. е. не меняться при замене одного аргумента любым другим и наоборот. Отсюда, в частности, вытекает, что

  10.3.  Applications of theorems on numerical characteristics .

Вместе с тем нам известно, что в сумме случайные величины   10.3.  Applications of theorems on numerical characteristics образуют единицу, следовательно, по теореме сложения математических ожиданий,

  10.3.  Applications of theorems on numerical characteristics ,

from where

  10.3.  Applications of theorems on numerical characteristics .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis