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

Lecture



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   12.1.  The distribution law of the monotone function of one random argument with distribution density   12.1.  The distribution law of the monotone function of one random argument . Other random variable   12.1.  The distribution law of the monotone function of one random argument associated with her functional dependency:

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

It is required to find the distribution density   12.1.  The distribution law of the monotone function of one random argument .

Consider the plot of the x-axis   12.1.  The distribution law of the monotone function of one random argument containing all possible values   12.1.  The distribution law of the monotone function of one random argument i.e.

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

In the particular case when the range of possible values   12.1.  The distribution law of the monotone function of one random argument unlimited   12.1.  The distribution law of the monotone function of one random argument ,   12.1.  The distribution law of the monotone function of one random argument .

The way to solve the problem depends on the behavior of the function   12.1.  The distribution law of the monotone function of one random argument Location on   12.1.  The distribution law of the monotone function of one random argument : whether it increases on this site or decreases, or fluctuates.

In this   12.1.  The distribution law of the monotone function of one random argument we consider the case when the function   12.1.  The distribution law of the monotone function of one random argument plot   12.1.  The distribution law of the monotone function of one random argument monotone. At the same time, we will analyze two cases separately: a monotone increase and a monotone decrease of a function.

1. Function   12.1.  The distribution law of the monotone function of one random argument Location on   12.1.  The distribution law of the monotone function of one random argument monotonously increases (fig. 12.1.1). When the magnitude   12.1.  The distribution law of the monotone function of one random argument takes on different values ​​on the plot   12.1.  The distribution law of the monotone function of one random argument random point   12.1.  The distribution law of the monotone function of one random argument moves only along the curve   12.1.  The distribution law of the monotone function of one random argument ; the ordinate of this random point is completely determined by its abscissa.

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

Fig. 12.1.1.

Denote   12.1.  The distribution law of the monotone function of one random argument distribution density   12.1.  The distribution law of the monotone function of one random argument . In order to determine   12.1.  The distribution law of the monotone function of one random argument first find the distribution function of   12.1.  The distribution law of the monotone function of one random argument :

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

Draw a straight line   12.1.  The distribution law of the monotone function of one random argument parallel to the x-axis at a distance   12.1.  The distribution law of the monotone function of one random argument from her (Fig. 12.1.1). To satisfy the condition   12.1.  The distribution law of the monotone function of one random argument random point   12.1.  The distribution law of the monotone function of one random argument should get to that part of the curve that lies below the straight   12.1.  The distribution law of the monotone function of one random argument ; for this it is necessary and sufficient that the random variable   12.1.  The distribution law of the monotone function of one random argument got to the plot of the x-axis from   12.1.  The distribution law of the monotone function of one random argument before   12.1.  The distribution law of the monotone function of one random argument where   12.1.  The distribution law of the monotone function of one random argument - abscissa of the intersection point of the curve   12.1.  The distribution law of the monotone function of one random argument and straight   12.1.  The distribution law of the monotone function of one random argument . Consequently,

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

Upper limit of the integral   12.1.  The distribution law of the monotone function of one random argument can be expressed through   12.1.  The distribution law of the monotone function of one random argument :

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

Where   12.1.  The distribution law of the monotone function of one random argument - inverse function   12.1.  The distribution law of the monotone function of one random argument . Then

  12.1.  The distribution law of the monotone function of one random argument . (12.1.1)

Differentiating the integral (12.1.1) by variable   12.1.  The distribution law of the monotone function of one random argument entering the upper limit, we get:

  12.1.  The distribution law of the monotone function of one random argument . (12.1.2)

2. Function   12.1.  The distribution law of the monotone function of one random argument Location on   12.1.  The distribution law of the monotone function of one random argument monotonously decreases (fig. 12.1.2).

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

Fig. 12.1.2.

In this case

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

from where

  12.1.  The distribution law of the monotone function of one random argument . (12.1.3)

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

  12.1.  The distribution law of the monotone function of one random argument . (12.1.4)

Indeed, when   12.1.  The distribution law of the monotone function of one random argument increases, its derivative (and hence   12.1.  The distribution law of the monotone function of one random argument ) is positive. With decreasing function   12.1.  The distribution law of the monotone function of one random argument derivative   12.1.  The distribution law of the monotone function of one random argument 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   12.1.  The distribution law of the monotone function of one random argument Subject to the Cauchy law with a distribution density:

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

Magnitude   12.1.  The distribution law of the monotone function of one random argument associated with   12.1.  The distribution law of the monotone function of one random argument addiction

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

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

Decision. Since the function   12.1.  The distribution law of the monotone function of one random argument monotone on the plot   12.1.  The distribution law of the monotone function of one random argument , 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:

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

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

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

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

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

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

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

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

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

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

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

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


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis