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

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

at any .

On the contrary, in case depends on then

.

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

Indeed let does not depend on :

. (8.5.1)

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

,

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

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

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

, (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 it can be concluded that random variables , are independent, namely if the distribution density splits into the product of two functions, of which one depends only on the other is only from , then the random variables are independent.

Example. System density has the form:

.

Determine whether random variables are dependent or independent. and .

Decision. Decomposing the denominator into factors, we have:

.

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

;

similarly

,

how do we make sure that

and therefore the values and 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 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 and 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 and 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 is related to the value probabilistic dependence, then, knowing the value , you can not specify the exact value , and you can specify only its distribution law, depending on what value the value took .

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 and are in probabilistic dependence, it does not mean that with a change in the magnitude magnitude changes in a very specific way; it only means that with changing magnitude magnitude also tends to change (for example, to increase or decrease with increasing ). 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: - growth at random taken person - its weight. Obviously the magnitude and 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:

.

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: - growth at random taken person; - his age. Obviously, for an adult size and can be considered practically independent; on the contrary, for a child of magnitude and 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 - the weight of the stone; random value - the greatest length of the stone. Values and are explicitly probabilistic dependence.

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

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

In the next 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.