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10.1. Expectation function. Function dispersion

Lecture



In solving various problems associated with random phenomena, modern probability theory makes extensive use of the apparatus of random variables. In order to use this apparatus, it is necessary to know the laws of the distribution of random majesty appearing in the problem. Generally speaking, these laws can be determined from experience, but usually experience whose goal is to determine the law of distribution of a random variable or a system of random variables (especially in the field of military equipment) turns out to be both difficult and expensive. Naturally, the problem arises - to reduce the amount of experiment to a minimum and make a judgment about the laws of distribution of random variables indirectly, based on the already known laws of distribution of other random variables. Such indirect methods of research of random variables play a very large role in the theory of probability. In this case, the random variable that is usually of interest to us is represented as a function of other random variables; knowing the laws of the distribution of arguments, it is often possible to establish the law of distribution of a function. We will meet with a number of tasks of this type in the future (see Chapter 12).

However, in practice, there are often cases when there is no special need to fully determine the distribution law of the function of random variables, and it suffices only to indicate its numerical characteristics: expectation, variance, sometimes some of the highest moments. In addition, very often the very laws of the distribution of arguments are not well known. In this connection, the problem often arises of determining only numerical characteristics of functions of random variables.

Consider the following problem: a random variable   10.1.  Expectation function.  Function dispersion there is a function of several random variables   10.1.  Expectation function.  Function dispersion :

  10.1.  Expectation function.  Function dispersion .

Let us know the law of distribution of the system of arguments.   10.1.  Expectation function.  Function dispersion ; required to find the numerical characteristics of the value   10.1.  Expectation function.  Function dispersion in the first place - the expectation and variance.

Imagine that we managed in one way or another to find the distribution law   10.1.  Expectation function.  Function dispersion magnitudes   10.1.  Expectation function.  Function dispersion . Then the problem of determining numerical characteristics becomes trivial; they are according to the formulas:

  10.1.  Expectation function.  Function dispersion ;

  10.1.  Expectation function.  Function dispersion

etc.

However, the very task of finding the distribution law   10.1.  Expectation function.  Function dispersion magnitudes   10.1.  Expectation function.  Function dispersion often turns out to be quite complicated. In addition, to solve the problem posed by us, finding the distribution law for the quantity   10.1.  Expectation function.  Function dispersion as such, it is not at all necessary: ​​to find only numerical characteristics of a quantity   10.1.  Expectation function.  Function dispersion there is no need to know its distribution law; enough to know the distribution of arguments   10.1.  Expectation function.  Function dispersion . Moreover, in some cases, in order to find the numerical characteristics of a function, it is not even necessary to know the distribution law of its arguments; it is enough to know only some numerical characteristics of the arguments.

Thus, the problem arises of determining the numerical characteristics of functions of random variables in addition to the laws of distribution of these functions.

Consider the problem of determining the numerical characteristics of a function for a given distribution of arguments. Let's start with the simplest case - the function of one argument - and set the following task.

There is a random variable   10.1.  Expectation function.  Function dispersion with a given distribution law; another random variable   10.1.  Expectation function.  Function dispersion associated with   10.1.  Expectation function.  Function dispersion functional dependence:

  10.1.  Expectation function.  Function dispersion .

Required without finding the distribution law of magnitude   10.1.  Expectation function.  Function dispersion , determine its expectation:

  10.1.  Expectation function.  Function dispersion . (10.1.1)

Consider first the case when   10.1.  Expectation function.  Function dispersion there is a discontinuous random variable with a series of distribution:

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

Write down possible values   10.1.  Expectation function.  Function dispersion and the probabilities of these values:

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion

(10.1.2)

Table (10.1.2) is not strictly a series of distribution of   10.1.  Expectation function.  Function dispersion , since in general some of the values

  10.1.  Expectation function.  Function dispersion (10.1.3)

may coincide with each other; moreover, these values ​​in the upper column of the table (10.1.2) do not necessarily go in ascending order. In order to move from a table (10.1.2) to a genuine series of distribution of   10.1.  Expectation function.  Function dispersion , it would be necessary to arrange the values ​​(10.1.3) in ascending order, combine the columns corresponding to the equal values   10.1.  Expectation function.  Function dispersion and add up the corresponding probabilities. But in this case we are not interested in the distribution law   10.1.  Expectation function.  Function dispersion as such; for our purposes — determining the expectation — it is enough to have such an “unordered” form of the distribution series as (10.1.2). Expectation value   10.1.  Expectation function.  Function dispersion can be determined by the formula

  10.1.  Expectation function.  Function dispersion (10.1.4)

Obviously the magnitude   10.1.  Expectation function.  Function dispersion , determined by the formula (10.1.4), cannot change due to the fact that under the sign of the sum some members will be combined in advance, and the order of the members will be changed.

The formula (10.1.4) for the mathematical expectation of a function is not explicitly contained in the distribution law of the function itself, but only the distribution law of the argument. Thus, to determine the mathematical expectation of a function, it is not at all necessary to know the distribution law of this function, but it is sufficient to know the distribution law of the argument.

Replacing in the formula (10.1.4) the sum with the integral, and the probability   10.1.  Expectation function.  Function dispersion - an element of probability, we obtain a similar formula for a continuous random variable:

  10.1.  Expectation function.  Function dispersion . (10.1.5)

Where   10.1.  Expectation function.  Function dispersion - distribution density   10.1.  Expectation function.  Function dispersion .

Similarly, the expectation of a function can be determined   10.1.  Expectation function.  Function dispersion from two random arguments   10.1.  Expectation function.  Function dispersion and   10.1.  Expectation function.  Function dispersion . For discontinuous values

  10.1.  Expectation function.  Function dispersion , (10.1.6)

Where   10.1.  Expectation function.  Function dispersion - the probability that the system   10.1.  Expectation function.  Function dispersion will take values   10.1.  Expectation function.  Function dispersion .

For continuous values

  10.1.  Expectation function.  Function dispersion , (10.1.7)

Where   10.1.  Expectation function.  Function dispersion - system distribution density   10.1.  Expectation function.  Function dispersion .

The mathematical expectation of a function is determined in exactly the same way from an arbitrary number of random arguments. We present the corresponding formula only for continuous values:

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion (10.1.8)

Where   10.1.  Expectation function.  Function dispersion - system distribution density   10.1.  Expectation function.  Function dispersion .

Formulas of the type (10.1.8) are very often encountered in the practical application of probability theory when it comes to averaging some values ​​depending on a number of random arguments.

Thus, the expectation of a function of any number of random arguments can be found in addition to the law of distribution of the function. Similarly, other numerical characteristics of the function can be found - moments of different orders. Since each moment is the mathematical expectation of a certain function of the random variable under study, the calculation of any moment can be carried out by methods completely similar to the above. Here we present calculation formulas only for variance, and only for the case of continuous random arguments.

The variance of a function of one random argument is expressed by the formula

  10.1.  Expectation function.  Function dispersion , (10.1.9)

Where   10.1.  Expectation function.  Function dispersion - expectation function   10.1.  Expectation function.  Function dispersion ;   10.1.  Expectation function.  Function dispersion - distribution density   10.1.  Expectation function.  Function dispersion .

The variance of the function of two arguments is expressed similarly:

  10.1.  Expectation function.  Function dispersion (10.1.10)

Where   10.1.  Expectation function.  Function dispersion - expectation function   10.1.  Expectation function.  Function dispersion ;   10.1.  Expectation function.  Function dispersion - system distribution density   10.1.  Expectation function.  Function dispersion .

Finally, in the case of an arbitrary number of arguments, in a similar notation:

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion . (10.1.11)

Note that often when calculating the variance it is convenient to use the ratio between the initial and central moments of the second order (see Chapter 5) and write:

  10.1.  Expectation function.  Function dispersion ; (10.1.12)

  10.1.  Expectation function.  Function dispersion ; (10.1.13)

  10.1.  Expectation function.  Function dispersion

  10.1.  Expectation function.  Function dispersion . (10.1.14)

Formulas (10.1.12) - (10.1.14) can be recommended when they do not lead to differences of close numbers, i.e. when   10.1.  Expectation function.  Function dispersion relatively small.

Let us consider several examples illustrating the application of the above methods for solving practical problems.

Example 1. A length segment is specified on a plane.   10.1.  Expectation function.  Function dispersion (fig. 10.1.1), rotating randomly so that all directions are equally likely. The segment is projected on a fixed axis.   10.1.  Expectation function.  Function dispersion . Determine the average length of the projection of the segment.

  10.1.  Expectation function.  Function dispersion

Fig. 10.1.1

Decision. The length of the projection is:

  10.1.  Expectation function.  Function dispersion ,

where is the angle   10.1.  Expectation function.  Function dispersion - random variable distributed with uniform density on the plot   10.1.  Expectation function.  Function dispersion .

By the formula (10.1.5) we have:

  10.1.  Expectation function.  Function dispersion .

Example 2. The elongated fragment of the projectile, which can be schematically depicted a segment of length   10.1.  Expectation function.  Function dispersion , flies, rotating around the center of mass in such a way that all its orientations in space are equally likely. On its way, the fragment meets a flat screen, perpendicular to the direction of its movement, and leaves a hole in it. Find the expectation of the length of this hole.

Decision. First of all, we give the mathematical formulation of the assertion that “all orientations of a fragment in space are equally probable.” Cut direction   10.1.  Expectation function.  Function dispersion we will characterize the unit vector   10.1.  Expectation function.  Function dispersion (fig. 10.1.2).

  10.1.  Expectation function.  Function dispersion

Fig. 10.1.2

Vector direction   10.1.  Expectation function.  Function dispersion in a spherical coordinate system associated with a plane   10.1.  Expectation function.  Function dispersion on which the design is made, is determined by two angles:   10.1.  Expectation function.  Function dispersion lying in a plane   10.1.  Expectation function.  Function dispersion and angle   10.1.  Expectation function.  Function dispersion lying in a plane perpendicular to   10.1.  Expectation function.  Function dispersion . With equal probability of all directions of the vector   10.1.  Expectation function.  Function dispersion all positions of its end on the surface of a sphere of a single radius   10.1.  Expectation function.  Function dispersion must have the same probability density; therefore, the element of probability

  10.1.  Expectation function.  Function dispersion ,

Where   10.1.  Expectation function.  Function dispersion - density of distribution of angles   10.1.  Expectation function.  Function dispersion , must be proportional to the elemental area   10.1.  Expectation function.  Function dispersion on the sphere   10.1.  Expectation function.  Function dispersion ; this elementary area is equal

  10.1.  Expectation function.  Function dispersion ,

from where

  10.1.  Expectation function.  Function dispersion ;   10.1.  Expectation function.  Function dispersion ,

Where   10.1.  Expectation function.  Function dispersion - coefficient of proportionality.

Coefficient value   10.1.  Expectation function.  Function dispersion we find from the relation

  10.1.  Expectation function.  Function dispersion ,

from where

  10.1.  Expectation function.  Function dispersion .

Thus, the density of the distribution of angles   10.1.  Expectation function.  Function dispersion expressed by the formula

  10.1.  Expectation function.  Function dispersion at   10.1.  Expectation function.  Function dispersion (10.1.15)

Design the segment na plane   10.1.  Expectation function.  Function dispersion ; projection length is equal to:

  10.1.  Expectation function.  Function dispersion .

Considering   10.1.  Expectation function.  Function dispersion as a function of two arguments   10.1.  Expectation function.  Function dispersion and   10.1.  Expectation function.  Function dispersion and applying the formula (10.1.7), we obtain:

  10.1.  Expectation function.  Function dispersion .

Thus, the average length of a hole left by a fragment in the screen is equal to   10.1.  Expectation function.  Function dispersion shard lengths

Example 3. Flat area figure   10.1.  Expectation function.  Function dispersion randomly rotates in space so that all orientations of this figure are equally likely. Find the average projected area of ​​the figure   10.1.  Expectation function.  Function dispersion on a fixed plane   10.1.  Expectation function.  Function dispersion (fig. 10.1.3).

  10.1.  Expectation function.  Function dispersion

Fig. 10.1.3

Decision. Direction of the plane of the figure   10.1.  Expectation function.  Function dispersion in space we will characterize the direction of the normal   10.1.  Expectation function.  Function dispersion to this plane. With a plane   10.1.  Expectation function.  Function dispersion bind the same spherical coordinate system as in the previous example. Normal direction   10.1.  Expectation function.  Function dispersion to the site   10.1.  Expectation function.  Function dispersion characterized by random angles   10.1.  Expectation function.  Function dispersion and   10.1.  Expectation function.  Function dispersion distributed with density (10.1.5). Square   10.1.  Expectation function.  Function dispersion figure projections   10.1.  Expectation function.  Function dispersion on the plane   10.1.  Expectation function.  Function dispersion equals

  10.1.  Expectation function.  Function dispersion ,

and the average projection area

  10.1.  Expectation function.  Function dispersion .

Thus, the average projected area of ​​an arbitrarily oriented flat figure on a fixed plane is equal to half the area of ​​this figure.

Example 4. In the process of tracking by a radar behind a certain object, the spot depicting the object is kept all the time within the limits of the screen. The screen is a circle   10.1.  Expectation function.  Function dispersion radius   10.1.  Expectation function.  Function dispersion . The spot occupies a random position on the screen with a constant probability density. Find the average distance from the spot to the center of the screen.

Decision. Denoting the distance   10.1.  Expectation function.  Function dispersion we have   10.1.  Expectation function.  Function dispersion where   10.1.  Expectation function.  Function dispersion - coordinates of the spot;   10.1.  Expectation function.  Function dispersion within a circle   10.1.  Expectation function.  Function dispersion and is zero beyond. Applying the formula (10.1.7) and passing to the polar coordinates in the integral, we have:

  10.1.  Expectation function.  Function dispersion .

Example 5. Reliability (probability of failure-free operation) of a technical device is a certain function.   10.1.  Expectation function.  Function dispersion three parameters characterizing the operation of the regulator. Options   10.1.  Expectation function.  Function dispersion are random variables with a known distribution density   10.1.  Expectation function.  Function dispersion . Find the average value (mathematical expectation) of the reliability of the device and the standard deviation characterizing its stability.

Decision. Device reliability   10.1.  Expectation function.  Function dispersion there is a function of three random variables (parameters)   10.1.  Expectation function.  Function dispersion . Its average value (mathematical expectation) can be found by the formula (10.1.8):

  10.1.  Expectation function.  Function dispersion . (10.1.16)

According to the formula (10.1.14) we have:

  10.1.  Expectation function.  Function dispersion ,

  10.1.  Expectation function.  Function dispersion .

Formula (10.1.16), expressing the average (full) probability of failure-free operation of the device, taking into account the random greatness, on which this probability depends in each particular case, is a special case of the so-called integral formula of total probability, which generalizes the usual total probability formula to the case of infinite (uncountable) number of hypotheses.

We derive here this formula in general.

Suppose that an experience in which an event of interest may or may not appear   10.1.  Expectation function.  Function dispersion , proceeds in random, previously unknown conditions. Let these conditions be characterized by continuous random variables.

  10.1.  Expectation function.  Function dispersion , (10.1.17)

distribution density of which

  10.1.  Expectation function.  Function dispersion .

Probability   10.1.  Expectation function.  Function dispersion occurrence of an event   10.1.  Expectation function.  Function dispersion there is a certain function of random variables (10.1.17):

  10.1.  Expectation function.  Function dispersion . (10.1.18)

We need to find the mean value of this probability or, in other words, the total probability of an event.   10.1.  Expectation function.  Function dispersion :

  10.1.  Expectation function.  Function dispersion .

Applying formula (10.1.8) for the mathematical expectation of a function, we find:

  10.1.  Expectation function.  Function dispersion . (10.1.19)

Формула (10.1.19) называется интегральной формулой полной вероятности. Нетрудно заметить, что по своей структуре она сходна с формулой полной вероятности, если заменить дискретный ряд гипотез непрерывной гаммой, сумму - интегралом, вероятность гипотезы - элементом вероятности:

  10.1.  Expectation function.  Function dispersion ,

а условную вероятность события при данной гипотезе - условной вероятностью события при фиксированных значениях случайных величин:

  10.1.  Expectation function.  Function dispersion .

Не менее часто, чем интегральной формулой полной вероятности пользуются интегральной формулой полного математического ожидания. Эта формула выражает среднее (полное) математическое ожидание случайной величины   10.1.  Expectation function.  Function dispersion , значение которой принимается в опыте, условия которого заранее неизвестны (случайны). Если эти условия характеризуются непрерывными случайными величинами

  10.1.  Expectation function.  Function dispersion

with distribution density

  10.1.  Expectation function.  Function dispersion ,

and the expectation of a quantity   10.1.  Expectation function.  Function dispersion is a function of  10.1.  Expectation function.  Function dispersion :

  10.1.  Expectation function.  Function dispersion ,

then the total expected value   10.1.  Expectation function.  Function dispersion calculated by the formula

  10.1.  Expectation function.  Function dispersion (10.1.20)

which is called the integral total expectation formula.

Example 6. The mathematical expectation of the distance   10.1.  Expectation function.  Function dispersion at which an object will be detected using four radar stations depends on some of the technical parameters of these stations:

  10.1.  Expectation function.  Function dispersion ,

which are independent random variables with distribution density

  10.1.  Expectation function.  Function dispersion .

For fixed parameters, the   10.1.  Expectation function.  Function dispersion expectation of the detection range is

  10.1.  Expectation function.  Function dispersion .

Find the average (full) expectation of the detection range.

Decision. According to the formula (10.1.20) we have:

  10.1.  Expectation function.  Function dispersion .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis