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

Lecture



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.   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution with distribution density   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . Consider the sum of random variables.   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution :

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

and find the distribution law   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . To do this, build on the plane   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution line whose equation   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution (fig. 12.5.1). This is a straight line that cuts off segments on the axes that are equal to   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . Straight   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution divides the plane   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution into two parts; to the right and above her   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ; to the left and below   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . Region   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution in this case, the lower left part of the plane   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution shaded in fig. 12.5.1. According to the formula (12.4.2) we have:

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

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

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

Fig. 12.5.1.

Differentiating this expression by variable   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution entering into the upper limit of the internal integral, we obtain:

  12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . (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   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution You can write another version of the same formula:

  12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution , (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   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution 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.   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution subordinate according to the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ; it is required to produce a composition of these laws, i.e., to find the density distribution of the quantity

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

As magnitudes   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution independent then

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

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

  12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution , (12.5.3)

  12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . (12.5.4)

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

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

Where   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution - 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   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution (or at least one of them) are given by one formula for the whole range of argument values ​​(from   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution before   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ). 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   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution 12.4, i.e. calculate the distribution function   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution magnitudes   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and differentiate this function.

Example 1. Compose the composition of a normal law:

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

and the law of uniform density:

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

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

  12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . (12.5.5)

The integrand function in expression (12.5.5) is nothing more than a normal law with a scattering center   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and standard deviation   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution , and the integral in expression (12.5.5) is the probability that a random variable subject to this law will hit a plot from   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution before   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ; Consequently,

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

Law Charts   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ,   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ,   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ,   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ,   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution are shown in fig. 12.5.2.

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

Fig. 12.5.2.

Example 2. Compose two laws of uniform density, given in the same area   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution :

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

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

Decision. Since the laws   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution only specified on certain axes   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution and   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution , 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   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution 12.4, and find the distribution function   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution magnitudes   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution .

Consider a random point   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution on surface   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . The area of ​​its possible positions is a square.   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution with a side equal to 1 (Fig. 12.5.3).

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

Fig. 12.5.3.

We have

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

where is the area   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution - part of the square   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution lying to the left and below the straight line   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution . Obviously

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

Where   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution - area of ​​the region   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution .

Let's make expression for the area of ​​area   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution at different values   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution taking advantage of rice 12.5.3 and 12.5.4:

1) at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ;

2) at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ;

3) at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ;

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

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

Fig. 12.5.4.

Differentiating these expressions, we get

1) at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ;

2) at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ;

3) at   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution ;

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

Distribution law   12.5.  The distribution law of the sum of two random variables.  The composition of the laws of distribution given in fig. 12.5.5. Such a law is called "Simpson's Law" or "Triangle Law".

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

Fig. 12.5.5.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis