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5.4. Distribution density

Lecture



Let there be a continuous random variable   5.4.  Distribution density with distribution function   5.4.  Distribution density which we assume is continuous and differentiable. Calculate the probability of hitting this random variable on the area from   5.4.  Distribution density before   5.4.  Distribution density :

  5.4.  Distribution density ,

those. the increment of the distribution function in this area. Consider the ratio of this probability to the length of the segment, i.e. average probability per unit length in this area, and we will approximate   5.4.  Distribution density to zero. In the limit, we obtain the derivative of the distribution function:

  5.4.  Distribution density . (5.4.1)

We introduce the notation:

  5.4.  Distribution density . (5.4.2)

Function   5.4.  Distribution density - the derivative of the distribution function - characterizes, as it were, the density with which the values ​​of a random variable at a given point are distributed. This function is called the distribution density (otherwise - "probability density") of a continuous random variable.   5.4.  Distribution density .

The terms “distribution density”, “probability density” become especially vivid when using the mechanical interpretation of a distribution; in this interpretation the function   5.4.  Distribution density literally characterizes the density of mass distribution along the abscissa axis (the so-called "linear density"). The curve depicting the density distribution of a random variable is called the distribution curve (Fig. 5.4.1).

  5.4.  Distribution density

Fig. 5.4.1.

Distribution density, as well as the distribution function, is one of the forms of the distribution law. In contrast to the distribution function, this form is not universal: it exists only for continuous random variables.

Consider a continuous random variable   5.4.  Distribution density with distribution density   5.4.  Distribution density and elementary plot   5.4.  Distribution density adjacent to the point   5.4.  Distribution density (Fig. 5.4.2). Probability of hitting a random variable   5.4.  Distribution density on this elementary segment (up to infinitesimal higher order) is equal   5.4.  Distribution density . Magnitude   5.4.  Distribution density called the element of probability. Geometrically, this is the area of ​​an elementary rectangle based on a segment   5.4.  Distribution density (Fig. 5.4.2).

  5.4.  Distribution density

Fig. 5.4.2.

Express the probability of hitting the value   5.4.  Distribution density on the segment from   5.4.  Distribution density before   5.4.  Distribution density (Figure 5.4.3) through the distribution density. Obviously, it is equal to the sum of the elements of probability throughout this segment, i.e. integral:

  5.4.  Distribution density (5.4.3)

*) Since the probability of any single value of a continuous random variable is zero, we can consider here a segment   5.4.  Distribution density , not including the left end, i.e. discarding the equal sign in   5.4.  Distribution density .

Geometrically probability of hitting the value   5.4.  Distribution density on the plot   5.4.  Distribution density equal to the area of ​​the distribution curve based on this area (Fig. 5.4.3.).

  5.4.  Distribution density

Fig. 5.4.3.

The formula (5.4.2.) Expresses the distribution density through the distribution function. Let us set the inverse problem: to express the distribution function in terms of density. By definition

  5.4.  Distribution density ,

whence by the formula (5.4.3) we have:

  5.4.  Distribution density . (5.4.4)

Geometrically   5.4.  Distribution density is nothing but the area of ​​the distribution curve, lying to the left of the point   5.4.  Distribution density (Fig. 5.4.4).

  5.4.  Distribution density

Fig. 5.4.4.

We indicate the main properties of the density distribution.

1. The density of distribution is a non-negative function:

  5.4.  Distribution density .

This property directly follows from the fact that the distribution function   5.4.  Distribution density there is a non-decreasing function.

2. The integral in the infinite limits of the distribution density is equal to one:

  5.4.  Distribution density .

This follows from formula (5.4.4) and from the fact that   5.4.  Distribution density .

Geometrically, the basic properties of the distribution density mean that:

1) the entire distribution curve does not lie below the abscissa axis;

2) the total area bounded by the distribution curve and the abscissa axis is one.

Let us find out the dimension of the main characteristics of a random variable — the distribution function and the distribution density. Distribution function   5.4.  Distribution density , like any probability, there is a dimensionless quantity. Distribution density   5.4.  Distribution density , as can be seen from the formula (5.4.1), is inverse to the dimension of the random variable.

Example 1. The distribution function of a continuous random variable X is given by

  5.4.  Distribution density

a) Find the coefficient a.

b) Find the density of distribution   5.4.  Distribution density .

c) Find the probability of hitting the value   5.4.  Distribution density on a site from 0,25 to 0,5.

Decision. a) Since the distribution function of the magnitude   5.4.  Distribution density continuous then when   5.4.  Distribution density   5.4.  Distribution density from where   5.4.  Distribution density .

b) distribution density   5.4.  Distribution density expressed by the formula

  5.4.  Distribution density

c) According to the formula (5.3.1) we have:

  5.4.  Distribution density .

Example 2. Random variable   5.4.  Distribution density subject to the distribution law with a density of:

  5.4.  Distribution density at   5.4.  Distribution density

  5.4.  Distribution density at   5.4.  Distribution density or   5.4.  Distribution density .

a) Find the coefficient a.

b) Build a graph of distribution density   5.4.  Distribution density .

c) Find the distribution function   5.4.  Distribution density and build her schedule.

d) Find the probability of hitting the value.   5.4.  Distribution density on a plot from 0 to   5.4.  Distribution density .

Decision. a) To determine the coefficient a, we use the property of the density of distribution:

  5.4.  Distribution density ,

from where   5.4.  Distribution density .

b) density chart   5.4.  Distribution density presented in fig. 5.4.5.

  5.4.  Distribution density

Fig. 5.4.5.

c) According to the formula (5.4.4), we obtain the expression of the distribution function:

  5.4.  Distribution density

Function graph   5.4.  Distribution density shown in fig. 5.4.6.

  5.4.  Distribution density

Fig. 5.4.6.

g) According to the formula (5.3.1) we have:

  5.4.  Distribution density .

The same result, but in a somewhat more complicated way, can be obtained by the formula (5.4.3).

Example 3. The distribution density of a random variable   5.4.  Distribution density given by the formula:

  5.4.  Distribution density .

a) Build a density graph   5.4.  Distribution density .

b) Find the probability that the quantity   5.4.  Distribution density gets to the site (-1, +1).

Decision. a) The density graph is given in fig. 5.4.7.

  5.4.  Distribution density

Fig. 5.4.7.

b) According to the formula (5.4.3) we have:

  5.4.  Distribution density .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis