12.1. The distribution law of the monotone function of one random argument

In previous chapters, we introduced methods for determining the numerical characteristics of functions of random variables; The main convenience of these methods is that they do not require finding the laws of distribution of functions. However, sometimes it becomes necessary to determine not only numerical characteristics, but also the laws of the distribution of functions.

We begin by considering the most simple problem related to this class: the problem of the distribution of the function of one random argument. Since for practice continuous random variables are most important, we will solve the problem for them.

There is a continuous random variable with distribution density . Other random variable associated with her functional dependency:

.

It is required to find the distribution density .

Consider the plot of the x-axis containing all possible values i.e.

.

In the particular case when the range of possible values unlimited , .

The way to solve the problem depends on the behavior of the function Location on : whether it increases on this site or decreases, or fluctuates.

In this we consider the case when the function plot monotone. At the same time, we will analyze two cases separately: a monotone increase and a monotone decrease of a function.

1. Function Location on monotonously increases (fig. 12.1.1). When the magnitude takes on different values ​​on the plot random point moves only along the curve ; the ordinate of this random point is completely determined by its abscissa.

Fig. 12.1.1.

Denote distribution density . In order to determine first find the distribution function of :

.

Draw a straight line parallel to the x-axis at a distance from her (Fig. 12.1.1). To satisfy the condition random point should get to that part of the curve that lies below the straight ; for this it is necessary and sufficient that the random variable got to the plot of the x-axis from before where - abscissa of the intersection point of the curve and straight . Consequently,

.

Upper limit of the integral can be expressed through :

,

Where - inverse function . Then

. (12.1.1)

Differentiating the integral (12.1.1) by variable entering the upper limit, we get:

. (12.1.2)

2. Function Location on monotonously decreases (fig. 12.1.2).

Fig. 12.1.2.

In this case

,

from where

. (12.1.3)

Comparing formulas (12.1.2) and (12.1.3), we note that they can be combined into one:

. (12.1.4)

Indeed, when increases, its derivative (and hence ) is positive. With decreasing function derivative negative, but before it in the formula (12.1.3) there is a minus. Therefore, the formula (12.1.4), in which the derivative is taken in absolute value, is true in both cases. Thus, the problem of the distribution of the monotone function is solved.

Example. Random value Subject to the Cauchy law with a distribution density:

.

Magnitude associated with addiction

.

Find the density distribution .

Decision. Since the function monotone on the plot , you can apply the formula (12.1.4). We solve the problem solution in the form of two columns: in the left will be placed the notation for the functions adopted in the general solution of the problem, in the right - the specific functions corresponding to this example: