12.5. The distribution law of the sum of two random variables. The composition of the laws of distribution

We use the general method outlined above to solve one particular problem that is practical for practice, namely, to find the distribution law for the sum of two random variables.

There is a system of two random variables. with distribution density . Consider the sum of random variables. and :

,

and find the distribution law . To do this, build on the plane line whose equation (fig. 12.5.1). This is a straight line that cuts off segments on the axes that are equal to . Straight divides the plane into two parts; to the right and above her ; to the left and below . Region in this case, the lower left part of the plane shaded in fig. 12.5.1. According to the formula (12.4.2) we have:

.

Fig. 12.5.1.

Differentiating this expression by variable entering into the upper limit of the internal integral, we obtain:

. (12.5.1)

This is a general formula for the distribution density of the sum of two random variables.

For reasons of symmetry of the problem with respect to and You can write another version of the same formula:

, (12.5.2)

which is equivalent to the first and can be used instead.

Of particular practical importance is the case when the added random variables are independent. Then they talk about the composition of the laws of distribution.

To make the composition of two laws of distribution it means to find the law of distribution of the sum of two independent random variables subordinate to these laws of distribution.

We derive a formula for the composition of two laws of distribution. There are two independent random variables. and subordinate according to the laws of distribution and ; it is required to produce a composition of these laws, i.e., to find the density distribution of the quantity

.

As magnitudes and independent then

,

and formulas (12.5.1) and (12.5.2) take the form:

, (12.5.3)

. (12.5.4)

A symbolic notation is often used to denote the composition of the distribution laws

,

Where - the symbol of the composition.

Formulas (12.5.3) and (12.5.4) for the composition of the laws of distribution are convenient only when the laws of distribution and (or at least one of them) are given by one formula for the whole range of argument values ​​(from before ). If both laws are set at different sites by different equations (for example, two laws of uniform density), then it is more convenient to use the general method directly, described in 12.4, i.e. calculate the distribution function magnitudes and differentiate this function.

Example 1. Compose the composition of a normal law:

and the law of uniform density:

at .

Decision. Let us apply the formula of the composition of the laws of distribution in the form (12.5.4):

. (12.5.5)

The integrand function in expression (12.5.5) is nothing more than a normal law with a scattering center and standard deviation , and the integral in expression (12.5.5) is the probability that a random variable subject to this law will hit a plot from before ; Consequently,

.

Law Charts , and at , , , are shown in fig. 12.5.2.

Fig. 12.5.2.

Example 2. Compose two laws of uniform density, given in the same area :

at ,

at .

Decision. Since the laws and only specified on certain axes and , for solving this problem, it is more convenient to use not the formulas (12.5.3) and (12.5.4), but the general method outlined in 12.4, and find the distribution function magnitudes .

Consider a random point on surface . The area of ​​its possible positions is a square. with a side equal to 1 (Fig. 12.5.3).

Fig. 12.5.3.

We have

,

where is the area - part of the square lying to the left and below the straight line . Obviously

,

Where - area of ​​the region .

Let's make expression for the area of ​​area at different values taking advantage of rice 12.5.3 and 12.5.4:

1) at ;

2) at ;

3) at ;

4) at .

Fig. 12.5.4.

Differentiating these expressions, we get

1) at ;

2) at ;

3) at ;

4) at .

Distribution law given in fig. 12.5.5. Such a law is called "Simpson's Law" or "Triangle Law".

Fig. 12.5.5.