Are random variables independent. Applied math. Conditional laws of distribution. Regression

Dependents and Independents random variables

When studying systems of random variables, one should always pay attention to the degree and nature of their dependence. This dependence can be more or less pronounced, more or less close. In some cases, the relationship between random variables can be so close that, knowing the value of one random variable, you can accurately indicate the value of another. In the other extreme case, the dependence between random variables is so weak and remote that they can practically be considered independent.
The concept of independent random variables is one of the important concepts of probability theory.
A random variable $Y$ is said to be independent of the random variable $X$ if the law of distribution of the value $Y$ does not depend on the value of the value $X$.
For continuous random variables, the condition that $Y$ is independent of $X$ can be written as: $$f(y\mid x)=f_(2)(y)$$ for any $y$.
On the contrary, if $Y$ depends on $X$, then $$f(y\mid x) \neq f_(2)(y)$$ We prove that dependence or independence of random variables is always mutual: if the value $Y$ does not depend on $X$, then the value $X$ does not depend on $Y$.
Indeed, let $Y$ be independent of $X$: $$f(y\mid x)=f_(2)(y)$$ we have: $$f_(1)(x)f(y \mid x)=f_(2)(y)f(x\mid y)$$ whence, we get: $$f_(1)(x)=f(x\mid y)$$ which was to be proved.
Since the dependence and independence of random variables are always mutual, we can give a new definition of independent random variables.
Random variables $X$ and $Y$ are called independent if the law of distribution of each of them does not depend on the value of the other. Otherwise, the quantities $X$ and $Y$ are called dependent.
For independent continuous random variables, the distribution law multiplication theorem takes the form: $$f(x, y)=f_(1)(x)f_(2)(y)$$ i.e. the distribution density of a system of independent random variables is equal to the product of the densities distribution of individual quantities included in the system.
Often, by the very form of the function $f(x, y)$ one can conclude that the random variables $X, Y$ are independent, namely, if the distribution density $f(x, y)$ decomposes into the product two functions, one of which depends only on $x$, the other only on $y$, then the random variables are independent.
Example 1 The distribution density of the system $(X, Y)$ has the form: $$f(x, y)=\frac(1)(\pi ^(2)(x^(2)+y^(2)+x ^(2)y^(2)+1))$$ Determine whether the random variables $X$ and $Y$ are dependent or independent.
Solution. Factoring the denominator, we have: $$f(x, y)=\frac(1)(\pi (x^(2)+1))\frac(1)(\pi (y^(2)+1 ))$$ From the fact that the function $f(x, y)$ has split into a product of two functions, one of which depends only on $x$, and the other only on $y$, we conclude that the quantities $X$ and $Y$ must be independent. Indeed, applying the formulas, we have: $$f(x, y)=\frac(1)(\pi (x^(2)+1))\int_(-\infty)^(\infty)(\frac( dy)(\pi (y^(2)+1)))=\frac(1)(\pi (x^(2)+1))$$ similar to $$f(x, y)=(\frac (1)(\pi (y^(2)+1)))$$ whence we make sure that $$f(x, y)=f_(1)(x)f_(2)(y)$$ and, hence the quantities $X$ and $Y$ are independent.

Random events are called independent if the occurrence of one of them does not affect the probability of occurrence of other events.

Example 1 . If there are two or more urns with colored balls, then drawing any ball from one urn does not affect the probability of drawing other balls from the remaining urns.

For not dependent events fair probability multiplication theorem: probability joint(simultaneous)the occurrence of several independent random events is equal to the product of their probabilities:

P (A 1 and A 2 and A 3 ... and A k) \u003d P (A 1) ∙ P (A 2) ∙ ... ∙ P (A k). (7)

Joint (simultaneous) occurrence of events means that events occur and A 1 , and A 2 , and A 3… and And k.

Example 2 . There are two urns. One contains 2 black and 8 white balls, the other contains 6 black and 4 white. Let the event BUT- random selection of a white ball from the first urn, AT- from the second. What is the probability of choosing at random from these urns a white ball, i.e. what is equal to R (BUT and AT)?

Solution: probability of drawing a white ball from the first urn
R(BUT) = = 0.8 from the second – R(AT) = = 0.4. The probability of getting a white ball from both urns at the same time is
R(BUT and AT) = R(BUT)· R(AT) = 0,8∙ 0,4 = 0,32 = 32%.

Example 3 A reduced iodine diet causes thyroid enlargement in 60% of animals in a large population. For the experiment, 4 enlarged glands are needed. Find the probability that 4 randomly selected animals will have an enlarged thyroid gland.

Solution:Random event BUT- a random selection of an animal with an enlarged thyroid gland. According to the condition of the problem, the probability of this event R(BUT) = 0.6 = 60%. Then the probability of joint occurrence of four independent events- random selection of 4 animals with an enlarged thyroid gland - will be equal to:

R(BUT 1 and BUT 2 and BUT 3 and BUT 4) = 0,6 ∙ 0,6 ∙0,6 ∙ 0,6=(0,6) 4 ≈ 0,13 = 13%.

dependent events. Probability multiplication theorem for dependent events

Random events A and B are called dependent if the occurrence of one of them, for example, A changes the probability of occurrence of the other event - B. Therefore, two probability values are used for dependent events: unconditional and conditional probabilities .

If a BUT and AT dependent events, then the probability of the event occurring AT first (i.e. before the event BUT) is called unconditional probability of this event and is designated R(AT).Probability of the event AT provided that the event BUT already happened, is called conditional probability developments AT and denoted R(AT/BUT) or R A(AT).

The unconditional - R(BUT) and conditional - R(A/B) probabilities for the event BUT.

The probabilities multiplication theorem for two dependent events: the probability of the simultaneous occurrence of two dependent events A and B is equal to the product of the unconditional probability of the first event by the conditional probability of the second:

R(A and B)= P(BUT)∙P(B/A) , (8)

BUT, or

R(A and B)= P(AT)∙P(A/B), (9)

if the event occurs first AT.

Example 1. There are 3 black balls and 7 white balls in an urn. Find the probability that 2 white balls will be taken out of this urn one by one (and the first ball is not returned to the urn).

Solution: probability of drawing the first white ball (event BUT) is equal to 7/10. After it is taken out, 9 balls remain in the urn, 6 of which are white. Then the probability of the appearance of the second white ball (the event AT) is equal to R(AT/BUT) = 6/9, and the probability of getting two white balls in a row is

R(BUT and AT) = R(BUT)∙R(AT/BUT) = = 0,47 = 47%.

The given probability multiplication theorem for dependent events can be generalized to any number of events. In particular, for three events, bound friend with a friend:

R(BUT and AT and FROM)= P(BUT)∙ R(B/A)∙ R(C/AB). (10)

Example 2. In two kindergartens, each attended by 100 children, there was an outbreak of an infectious disease. The proportion of cases is 1/5 and 1/4, respectively, and in the first institution 70%, and in the second - 60% of cases are children under 3 years of age. One child is randomly selected. Determine the probability that:

1) the selected child belongs to the first kindergarten (event BUT) and sick (event AT).

2) a child is selected from the second kindergarten(event FROM), sick (event D) and older than 3 years (event E).

Solution. 1) the desired probability -

R(BUT and AT) = R(BUT) ∙ R(AT/BUT) = = 0,1 = 10%.

2) the desired probability:

R(FROM and D and E) = R(FROM) ∙ R(D/C) ∙ R(E/CD) = = 5%.

Bayes formula

= (12)

Example1. During the initial examination of the patient, 3 diagnoses are assumed H 1 , H 2 , H 3 . Their probabilities, according to the doctor, are distributed as follows: R(H 1) = 0,5; R(H 2) = 0,17; R(H 3) = 0.33. Therefore, the first diagnosis seems tentatively the most likely. To clarify it, for example, a blood test is prescribed, in which an increase in ESR is expected (event BUT). It is known in advance (based on research results) that the probabilities of an increase in ESR in suspected diseases are equal to:

R(BUT/H 1) = 0,1; R(BUT/H 2) = 0,2; R(BUT/H 3) = 0,9.

In the obtained analysis, an increase in ESR was recorded (event BUT happened). Then the calculation according to the Bayes formula (12) gives the values of the probabilities of the alleged diseases with an increased ESR value: R(H 1 /BUT) = 0,13; R(H 2 /BUT) = 0,09;
R(H 3 /BUT) = 0.78. These figures show that, taking into account laboratory data, not the first, but the third diagnosis, the probability of which has now turned out to be quite high, is the most realistic.

Example 2. Determine the probability that assesses the degree of risk of perinatal* death of a child in women with an anatomically narrow pelvis.

Solution: let event H 1 - safe delivery. According to clinical reports, R(H 1) = 0.975 = 97.5%, then if H 2- the fact of perinatal mortality, then R(H 2) = 1 – 0,975 = 0,025 = 2,5 %.

Denote BUT- the fact of the presence of a narrow pelvis in a woman in labor. From the studies carried out, it is known: a) R(BUT/H 1) - the probability of a narrow pelvis with favorable childbirth, R(BUT/H 1) = 0.029, b) R(BUT/H 2) - the probability of a narrow pelvis in perinatal mortality,
R(BUT/H 2) = 0.051. Then the desired probability of perinatal mortality in a narrow pelvis in a woman in labor is calculated by the Bays formula (12) and is equal to:

Thus, the risk of perinatal mortality in anatomically narrow pelvis is significantly higher (almost twice) than the average risk (4.4% vs. 2.5%).

Any of them does not depend on what values the other random variables have taken (or will take).

For example, the system of two playing dice - it is quite clear that the result of throwing one die does not affect the probabilities of the faces of another die falling out in any way. Or the same independently operating slot machines. And, probably, some have the impression that any SV is independent in general. However, this is not always the case.

Consider simultaneous discarding two magnet dice whose north poles are on the side of the 1-point face and the south poles are on the opposite 6-point face. Will similar random variables be independent? Yes, they will. The probabilities of dropping out "1" and "6" will simply decrease and the chances of other faces will increase, because as a result of the test, the cubes can be attracted by opposite poles.

Now consider a system in which the dice are discarded successively:

- the number of points rolled on the first die;

- the number of points rolled on the second die, provided that it is always discarded to the right (for example) side of the 1st die.

In this case, the distribution law of the random variable depends on how the 1st cube is located. The second bone can either be attracted, or vice versa - rebound (if the poles of the same name “met”), or partially or completely ignore the 1st cube.

Second example: suppose that the same slot machines are united in a single network, and - there is a system of random variables - winnings on the corresponding machines. I don’t know if this scheme is legal, but the owner of the gaming hall can easily set up the network in the following way: when a big win occurs on any machine, the laws of distribution of winnings on all machines automatically change. In particular, it is advisable to reset the probabilities of large winnings for a while so that the institution does not face a shortage of funds (in the event that suddenly someone wins big again). Thus, the considered system will be dependent.

As a demonstration example, consider a deck of 8 cards, let it be kings and queens, and a simple game in which two players consecutively (no matter in what order) draw one card from the deck. Consider a random variable , which symbolizes one player and takes the following values: 1 , if he drew a heart card, and 0 - if the card is of a different suit.

Similarly, let the random variable symbolize another player and also take the values 0 or 1 if he has drawn not a heart and a heart, respectively.

is the probability that both players will extract the worm,

is the probability of the opposite event, and:

- the probability that one will extract the worm, and the other - no; or vice versa:

Thus, the probability distribution law of the dependent system is:

Control: , which was to be verified. ...Perhaps you have a question, why am I considering exactly 8, and not 36 cards? Yes, just so that the fractions are not so cumbersome.

Now let's analyze the results a bit. If we sum the probabilities line by line: , then we get exactly the distribution law of the random variable :

It is easy to understand that this distribution corresponds to the situation when the "X" player draws a card alone, without a "G" comrade, and his expected value:
- is equal to the probability of extracting hearts from our deck.

Similarly, if we sum the probabilities by columns, then we obtain the law of distribution of a single game of the second player:

with the same expectation

Due to the "symmetry" of the rules of the game, the distributions turned out to be the same, but, in the general case, they are, of course, different.

In addition, it is useful to consider conditional laws of probability distribution . This is a situation where one of the random variables has already taken on a specific value, or we assume this hypothetically.

Let the "player" player draw a card first and draw not a heart. The probability of this event is (sum the probabilities over the first column tables - see above). Then, from the same multiplication theorems for the probabilities of dependent events we get the following conditional probabilities:
- the probability that the “X” player draws not a heart, provided that the “playing” player draws not a heart;
- the probability that the “X” player draws a heart, provided that the “player” player did not draw a heart.

... everyone remembers how to get rid of four-story fractions? And yes, formal but very comfortable technical rule for calculating these probabilities: first sum all probabilities by column, and then divide each probability by the resulting sum.

Thus, at , the conditional law of distribution of a random variable will be written as follows:

, OK. Let's calculate the conditional mathematical expectation:

Now let's draw up the law of distribution of a random variable under the condition that the random variable has taken the value , i.e. The "player" player drew a heart-suited card. To do this, we summarize the probabilities of the 2nd column tables ( see above): and calculate the conditional probabilities:
- the fact that the "X" player will draw not a worm,
- and a worm.
Thus, the desired conditional distribution law:

Control: , and conditional expectation:
- of course, it turned out to be less than in the previous case, since the "player" player reduced the number of hearts in the deck.

"Mirror" way (working with table rows) can be composed - the law of distribution of a random variable, provided that the random variable has taken the value , and conditional distribution, when the "X" player has taken the worm. It is easy to understand that due to the “symmetry” of the game, the same distributions and the same values will be obtained.

For continuous random variables introduce the same concepts. conditional distributions and mathematical expectations, but if there is no hot need for them, then it is better to continue studying this lesson.

In practice, in most cases, you will be offered a ready-made distribution law for a system of random variables:

Example 4

A two-dimensional random variable is given by its own probability distribution law:

... I wanted to consider a larger table, but I decided not to be manic, because the main thing is to understand the very principle of the solution.

Required:

1) Draw up distribution laws and calculate the corresponding mathematical expectations. Make a reasonable conclusion about the dependence or independence of random variables .

This is a task to solve on your own! I remind you that in the case of the independence of the NE, the laws must turn out to be the same and coincide with the law of distribution of a random variable , and the laws must coincide with . Decimals, who does not know or forgot, it is convenient to divide like this: .
You can check out the sample at the bottom of the page.

2) Calculate the coefficient of covariance.

First, let's look at the term itself, and where it came from at all: when a random variable takes on different values, then they say that it varies, and the quantitative measurement of this variations, as you know, is expressed dispersion. Using the formula for calculating the variance, as well as the properties of the expectation and variance, it is easy to establish that:

that is, when adding two random variables, their variances are summed up and an additional term is added that characterizes joint variation or shortly - covariance random variables.

covariance or correlation moment - this is measure of joint variation random variables.

Designation: or

The covariance of discrete random variables is defined, now I will “express” :), as the mathematical expectation of the product linear deviations of these random variables from the corresponding mathematical expectations:

If , then random variables dependent. Figuratively speaking, a non-zero value tells us about natural"responses" of one SW to a change in another SW.

Covariance can be calculated in two ways, I'll cover both.

Method one. By definition of mathematical expectation:

A "terrible" formula and not at all terrible calculations. First, we compose the laws of distribution of random variables and - for this we summarize the probabilities over the rows ("X" value) and by columns ("game" value):

Take a look at the original top table - does everyone understand how the distributions turned out? Compute expectations:
and deviations values of random variables from the corresponding mathematical expectations:

It is convenient to place the resulting deviations in a two-dimensional table, inside which then rewrite the probabilities from the original table:

Now you need to calculate all possible products, as an example, I highlighted: (Red color) and (Blue colour). It is convenient to carry out calculations in Excel, and write everything in detail on a clean copy. I’m used to working “line by line” from left to right, and therefore I will first list all possible products with an “X” deviation of -1.6, then with a deviation of 0.4:

Method two, simpler and more common. According to the formula:

The expectation of the product SW is defined as and technically everything is very simple: we take the original table of the problem and find all possible products by the corresponding probabilities ; in the figure below, I highlighted the work in red and blue product:

First, I will list all the products with the value , then with the value , but you, of course, can use a different order of enumeration - as you prefer:

The values have already been calculated (see Method 1), and it remains to apply the formula:

As noted above, the non-zero value of the covariance tells us about the dependence of random variables, and the more it is modulo, the more this dependence nearer to functional linear dependencies. For it is determined through linear deviations.

Thus, the definition can be formulated more precisely:

covariance is a measure linear dependencies of random variables.

With a value of zero, everything is more interesting. If it is established that , then the random variables may turn out to be both independent and dependent(because the dependence can be not only linear). In this way, this fact cannot generally be used to substantiate the independence of the SV!

However, if it is known that they are independent, then . This can be easily verified analytically: since for independent random variables the property ( see previous lesson), then according to the formula for calculating the covariance:

What values can this coefficient take? Covariance coefficient takes values not exceeding modulo- and the more, the more pronounced linear dependence. And everything seems to be fine, but there is a significant inconvenience of such a measure:

Suppose we explore two-dimensional continuous random variable(preparing mentally :)), the components of which are measured in centimeters, and received the value . By the way, what is the dimension of covariance? Since, - centimeters, and - also centimeters, then their product and the expectation of this product – expressed in square centimeters, i.e. covariance, like variance, is quadratic value.

Now suppose that someone learned the same system, but used not centimeters, but millimeters. Since 1 cm = 10 mm, the covariance will increase by 100 times and will be equal to !

Therefore, it is convenient to consider normalized a covariance coefficient that would give us the same and dimensionless value. This coefficient is called, we continue our task:

3) Coefficient correlations . Or, more precisely, the linear correlation coefficient:

, where - standard deviations random variables.

Correlation coefficient dimensionless and takes values from the range:

(if you have something else in practice - look for an error).

The more modulo to unity, the closer the linear relationship between the values, and the closer to zero, the less pronounced this dependence. The relationship is considered significant starting at about . The extreme values correspond to a strict functional dependence, but in practice, of course, there are no “ideal” cases.

I really want to give many interesting examples, but the correlation is more relevant in the course mathematical statistics and so I'll save them for the future. Well, now let's find the correlation coefficient in our problem. So. The laws of distribution are already known, I will copy from above:

Expectations are found: , and it remains to calculate the standard deviations. sign I won’t draw it up, it’s faster to calculate with the line:

Covariance found in the previous paragraph , and it remains to calculate the correlation coefficient:
, thus, between the values there is a linear dependence of the average tightness.

The fourth task is again more typical for tasks mathematical statistics, but just in case, consider it here:

4) Write a linear regression equation for .

The equation linear regression is a function , which the best way approximates the values of the random variable . For the best approximation, one usually uses least square method, and then the regression coefficients can be calculated by the formulas:
, these are miracles, and the 2nd coefficient:

Conditional laws of distribution. Regression.

Definition. The conditional distribution law of one of the one-dimensional components of a two-dimensional random variable (X, Y) is its distribution law, calculated under the condition that the other component took a certain value (or fell into some interval). In the previous lecture, finding conditional distributions for discrete random variables was considered. There are also formulas for conditional probabilities:

In the case of continuous random variables, it is necessary to determine the probability densities of the conditional distributions j y (x) and j X (y). To this end, in the above formulas, we will replace the probabilities of events with their "probability elements",!

after reduction by dx and dy we get:

those. the conditional probability density of one of the one-dimensional components of a two-dimensional random variable is equal to the ratio of its joint density to the probability density of the other component. These ratios are written in the form

are called the theorem (rule) of multiplication of distribution densities.

Conditional densities j y (x) and j X (y). have all the properties of "unconditional" density.

When studying two-dimensional random variables, we consider numerical characteristics one-dimensional components X and Y - mathematical expectations and variances. For a continuous random variable (X, Y), they are determined by the formulas:

Along with them, numerical characteristics of conditional distributions are also considered: conditional mathematical expectations M x (Y) and M y (X) and conditional variances D x (Y) and D Y (X). These characteristics are found by the usual formulas of mathematical expectation and variance, in which conditional probabilities or conditional probability densities are used instead of event probabilities or probability densities.

Conditional mathematical expectation of a random variable Y for X = x, i.e. M x (Y), there is a function of x, called the regression function or simply regression Y on X. Similarly, M Y (X) is called the regression function or simply regression X on Y. The graphs of these functions are called respectively regression lines (or regression curves) Y by X or X by Y.

Dependent and independent random variables.

Definition. Random variables X and Y are called independent if their joint distribution function F(x,y) is represented as a product of distribution functions F 1 (x) and F 2 (y) of these random variables, i.e.

Otherwise, random variables X and Y are called dependent.

Differentiating the equality twice with respect to the arguments x and y, we get

those. for independent continuous random variables X and Y, their joint density j(x, y) is equal to the product of the probability densities j 1 (x) and j 2 (y) of these random variables.

Until now, we have encountered the concept of a functional relationship between the variables X and Y, when each value of x in one variable corresponded to a strictly defined value in the other. For example, the relationship between two random variables - the number of failed pieces of equipment for certain period time and their cost - functional.

In general, one encounters a different type of dependency, less rigid than the functional dependency.

Definition. The relationship between two random variables is called probabilistic (stochastic or statistical) if each value of one of them corresponds to a certain (conditional) distribution of the other.

In the case of a probabilistic (stochastic) dependence, it is impossible, knowing the value of one of them, to accurately determine the value of the other, but you can only indicate the distribution of the other value. For example, the relationship between the number of equipment failures and the cost of its preventive maintenance, the weight and height of a person, the time spent by a schoolchild on watching television programs and reading books, etc. are probabilistic (stochastic).

On fig. 5.10 shows examples of dependent and independent random variables X and Y.

The concept of independent random variables is one of the important concepts of probability theory.

A random variable is called independent of a random variable if the distribution law of the value does not depend on what value the value has taken.

For continuous random variables, the condition of independence from can be written as:

for any .

On the contrary, if depends on , then

Let us prove that the dependence or independence of random variables is always mutual: if the value does not depend on .

Indeed, let it not depend on:

. (8.5.1)

From formulas (8.4.4) and (8.4.5) we have:

whence, taking into account (8.5.1), we obtain:

Q.E.D.

Since the dependence and independence of random variables are always mutual, it is possible to give a new definition of independent random variables.

Random variables and are called independent if the law of distribution of each of them does not depend on what value the other has taken. Otherwise, the quantities and are called dependent.

For independent continuous random variables, the distribution law multiplication theorem takes the form:

, (8.5.2)

i.e., the distribution density of a system of independent random variables is equal to the product of the distribution densities of individual variables included in the system.

Condition (8.5.2) can be regarded as a necessary and sufficient condition for the independence of random variables.

Often, by the very form of the function, one can conclude that the random variables are independent, namely, if the distribution density is divided into the product of two functions, one of which depends only on , the other only on , then the random variables are independent.

Example. The distribution density of the system has the form:

Determine whether the random variables and are dependent or independent.

Solution. Factoring the denominator, we have:

From the fact that the function is decomposed into the product of two functions, one of which depends only on and the other only on , we conclude that the quantities and must be independent. Indeed, applying formulas (8.4.2) and (8.4.3), we have:

;

likewise

how do we make sure that

and hence the quantities and are independent.

The above criterion for judging the dependence or independence of random variables is based on the assumption that we know the distribution law of the system. In practice, it often happens the other way around: the distribution law of the system is not known; only the laws of distribution of individual quantities included in the system are known, and there are grounds for believing that the quantities and are independent. Then it is possible to write the distribution density of the system as the product of the distribution densities of the individual quantities included in the system.

Let us dwell in more detail on the important concepts of "dependence" and "independence" of random variables.

The concept of "independence" of random variables, which we use in probability theory, is somewhat different from the usual concept of "dependence" of variables, which we operate in mathematics. Indeed, usually under the "dependence" of quantities they mean only one type of dependence - a complete, rigid, so-called - functional dependence. Two quantities and are called functionally dependent if, knowing the value of one of them, one can accurately indicate the value of the other.

In probability theory, we meet with another, more general, type of dependence - with a probabilistic or "stochastic" dependence. If the value is related to the value by a probabilistic dependence, then, knowing the value, it is impossible to specify the exact value of , but you can only indicate its distribution law, depending on what value the value has taken.

The probabilistic dependence may be more or less close; as the tightness of the probabilistic dependence increases, it approaches the functional one more and more. Thus, the functional dependence can be considered as an extreme, limiting case of the closest probabilistic dependence. Another extreme case is the complete independence of random variables. Between these two extreme cases lie all gradations of probabilistic dependence - from the strongest to the weakest. Those physical quantities, which in practice we consider to be functionally dependent, are actually connected by a very close probabilistic dependence: for a given value of one of these quantities, the other varies within such narrow limits that it can practically be considered quite definite. On the other hand, those quantities that we consider independent in practice and reality are often in some mutual dependence, but this dependence is so weak that it can be neglected for practical purposes.

Probabilistic dependence between random variables is very common in practice. If random variables and are in a probabilistic dependence, this does not mean that with a change in magnitude, the magnitude changes in a completely definite way; it just means that as the value changes, the value tends to change as well (for example, increase or decrease with increasing ). This trend is observed only "on average", in in general terms, and in each individual case, deviations from it are possible.

Consider, for example, two such random variables: - the height of a randomly taken person, - his weight. Obviously, the quantities and are in a certain probabilistic dependence; it is expressed in the fact that, in general, people with greater height have more weight. It is even possible to draw up an empirical formula that approximately replaces this probabilistic dependence with a functional one. Such, for example, is the well-known formula that approximately expresses the relationship between height and weight.

Are random variables independent. Applied math. Conditional laws of distribution. Regression

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