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8.5 Dependent and independent random variables

Lecture



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 specify the value of another. In the other extreme case, the relationship between random variables is so weak and distant that they can practically be considered independent.

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

Random value   8.5 Dependent and independent random variables called independent of random variable   8.5 Dependent and independent random variables if the distribution law   8.5 Dependent and independent random variables does not depend on what value the value took   8.5 Dependent and independent random variables .

For continuous random variables, the condition of independence   8.5 Dependent and independent random variables from   8.5 Dependent and independent random variables can be written as:

  8.5 Dependent and independent random variables

at any   8.5 Dependent and independent random variables .

On the contrary, in case   8.5 Dependent and independent random variables depends on   8.5 Dependent and independent random variables then

  8.5 Dependent and independent random variables .

We prove that the dependence or independence of random variables is always reciprocal: if the value   8.5 Dependent and independent random variables does not depend on   8.5 Dependent and independent random variables .

Indeed let   8.5 Dependent and independent random variables does not depend on   8.5 Dependent and independent random variables :

  8.5 Dependent and independent random variables . (8.5.1)

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

  8.5 Dependent and independent random variables ,

from which, taking into account (8.5.1), we obtain:

  8.5 Dependent and independent random variables

Q.E.D.

Since the dependence and independence of random variables are always reciprocal, a new definition of independent random variables can be given.

Random variables   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are called independent if the distribution law of each of them does not depend on what value the other takes. Otherwise the values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are called dependent.

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

  8.5 Dependent and independent random variables , (8.5.2)

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

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

Often by the very sight of the function   8.5 Dependent and independent random variables it can be concluded that random variables   8.5 Dependent and independent random variables ,   8.5 Dependent and independent random variables are independent, namely if the distribution density   8.5 Dependent and independent random variables splits into the product of two functions, of which one depends only on   8.5 Dependent and independent random variables the other is only from   8.5 Dependent and independent random variables , then the random variables are independent.

Example. System density   8.5 Dependent and independent random variables has the form:

  8.5 Dependent and independent random variables .

Determine whether random variables are dependent or independent.   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables .

Decision. Decomposing the denominator into factors, we have:

  8.5 Dependent and independent random variables .

From what function   8.5 Dependent and independent random variables broke up into a product of two functions, one of which is dependent only on   8.5 Dependent and independent random variables and the other is only from   8.5 Dependent and independent random variables , we conclude that the values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables must be independent. Indeed, applying formulas (8.4.2) and (8.4.3), we have:

  8.5 Dependent and independent random variables ;

similarly

  8.5 Dependent and independent random variables ,

how do we make sure that

  8.5 Dependent and independent random variables

and therefore the values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are independent.

The above criterion for judging the dependence or independence of random variables is based on the assumption that the distribution law of the system is known to us. In practice, the opposite is often the case: the law of distribution of a system   8.5 Dependent and independent random variables not known; only the laws of the distribution of individual quantities in the system are known, and there is reason to believe that the quantities   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are independent. Then you can write the distribution density of the system as the product of the distribution densities of the individual quantities 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 quantities, which we operate in mathematics. Indeed, usually by “dependency” of quantities one means only one type of dependency - a complete, rigid, so-called — functional dependency. Two values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables They are called functionally dependent if, knowing the value of one of them, you can accurately indicate the value of the other.

In the theory of probability, we encounter another, more general, type of dependence — with probabilistic or “stochastic” dependence. If the value   8.5 Dependent and independent random variables is related to the value   8.5 Dependent and independent random variables probabilistic dependence, then, knowing the value   8.5 Dependent and independent random variables , you can not specify the exact value   8.5 Dependent and independent random variables , and you can specify only its distribution law, depending on what value the value took   8.5 Dependent and independent random variables .

Probabilistic dependence may be more or less close; with increasing closeness of probabilistic dependence, it is increasingly approaching the functional one. 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 that we consider functionally dependent in practice are in fact closely related to a very close probabilistic dependence: for a given value of one of these quantities, the other fluctuates in such narrow limits that it can practically be considered quite definite. On the other hand, those values ​​that we consider independent in practice, and reality are often in some kind of interdependence, but this dependence is so weak that it can be neglected for practical purposes.

Probabilistic dependence between random variables is very often found in practice. If random values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are in probabilistic dependence, it does not mean that with a change in the magnitude   8.5 Dependent and independent random variables magnitude   8.5 Dependent and independent random variables changes in a very specific way; it only means that with changing magnitude   8.5 Dependent and independent random variables magnitude   8.5 Dependent and independent random variables also tends to change (for example, to increase or decrease with increasing   8.5 Dependent and independent random variables ). This tendency is observed only “on average”, in general, and in each individual case deviations from it are possible.

Consider, for example, two such random variables:   8.5 Dependent and independent random variables - growth at random taken person   8.5 Dependent and independent random variables - its weight. Obviously the magnitude   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are in a certain probability relationship; it is expressed in the fact that in general, people with greater growth have more weight. You can even make an empirical formula that approximately replaces this probabilistic dependence of the functional. Such, for example, is the well-known formula, approximately expressing the relationship between height and weight:

  8.5 Dependent and independent random variables .

Formulas of this type are obviously not exact and express only a certain average, mass regularity, a tendency from which deviations are possible in each individual case.

In the above example, we dealt with a case of explicit dependence. Now consider these two random variables:   8.5 Dependent and independent random variables - growth at random taken person;   8.5 Dependent and independent random variables - his age. Obviously, for an adult size   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables can be considered practically independent; on the contrary, for a child of magnitude   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are addicted.

Let us give some more examples of random variables that are in various degrees of dependence.

1. Of the stones that make up the pile of rubble, one stone is selected at random. Random value   8.5 Dependent and independent random variables - the weight of the stone; random value   8.5 Dependent and independent random variables - the greatest length of the stone. Values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables are explicitly probabilistic dependence.

2. Shooting a rocket at a predetermined area of ​​the ocean. Magnitude   8.5 Dependent and independent random variables - longitudinal error of the point of impact (undershoot, flight); random value   8.5 Dependent and independent random variables - error in the rocket speed at the end of the active part of the movement. Values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables obviously dependent because the error   8.5 Dependent and independent random variables is one of the main reasons for the longitudinal error   8.5 Dependent and independent random variables .

3. The aircraft, while in flight, measures the height above the Earth's surface using a barometric instrument. Two random variables are considered:   8.5 Dependent and independent random variables - height measurement error and   8.5 Dependent and independent random variables - weight of fuel remaining in the fuel tanks at the time of measurement. Values   8.5 Dependent and independent random variables and   8.5 Dependent and independent random variables can practically be considered independent.

In the next   8.5 Dependent and independent random variables we will get acquainted with some numerical characteristics of a system of random variables, which will enable us to estimate the degree of dependence of these quantities.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis