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5.8. Uniform density law

Lecture



In some practical tasks, there are continuously random variables, which are known in advance that their possible values ​​lie within a certain defined interval; in addition, it is known that within this interval, all values ​​of a random variable are equally likely (more precisely, they have the same probability distribution density). Such random variables are said to be distributed according to the law of uniform density.

We give several examples of such random variables.

Example 1. The body was weighed on an accurate scale, but at the weighing body there are only weights weighing at least 1g; the weighing result indicates that the body weight is between   5.8.  Uniform density law and   5.8.  Uniform density law grams. Body weight is assumed equal   5.8.  Uniform density law grams The mistake made at the same time   5.8.  Uniform density law obviously there is a random variable distributed with a uniform density on the plot   5.8.  Uniform density law year

Example 2. A vertical symmetrical wheel (fig. 5.8.1) is rotated and then stopped due to friction. Considered a random variable   5.8.  Uniform density law - the angle that, after stopping, will be with the horizon a fixed radius of the wheel OA. Obviously the magnitude   5.8.  Uniform density law distributed with uniform density on the plot   5.8.  Uniform density law .

  5.8.  Uniform density law

Fig. 5.8.1.

Example 3. Metro trains run at intervals of 2 minutes. The passenger goes to the platform at some point in time. The time T, during which he will have to wait for the train, is a random variable distributed with a uniform density over a stretch of (0.2) minutes.

  5.8.  Uniform density law

Fig. 5.8.2.

Consider a random variable   5.8.  Uniform density law subordinate to the law of uniform density in the area from   5.8.  Uniform density law before   5.8.  Uniform density law (fig. 5.8.2), and we write for it an expression of the distribution density   5.8.  Uniform density law . Density   5.8.  Uniform density law is constant and equal to c on the segment   5.8.  Uniform density law ; outside this segment, it is zero:

  5.8.  Uniform density law

Since the area bounded by the distribution curve is one:

  5.8.  Uniform density law

that

  5.8.  Uniform density law

and distribution density   5.8.  Uniform density law has the form:

  5.8.  Uniform density law (5.8.1)

Formula (5.8.1) and expresses the law of uniform density on the site   5.8.  Uniform density law .

We write an expression for the distribution function   5.8.  Uniform density law . The distribution function is expressed by the area of ​​the distribution curve to the left of the point.   5.8.  Uniform density law . Consequently,

  5.8.  Uniform density law

Function graph   5.8.  Uniform density law is shown in fig. 5.8.3.

  5.8.  Uniform density law

Fig. 5.8.3.

Define the basic numerical characteristics of a random variable.   5.8.  Uniform density law subordinate to the law of uniform density in the area from   5.8.  Uniform density law before   5.8.  Uniform density law .

Expectation value X:

  5.8.  Uniform density law (5.8.2)

Due to the symmetry of the uniform distribution, the median of   5.8.  Uniform density law also equal to   5.8.  Uniform density law .

Fashion law of uniformity of density is not.

According to the formula (5.7.16) we find the variance of the value   5.8.  Uniform density law :

  5.8.  Uniform density law (5.8.3)

where the standard deviation

  5.8.  Uniform density law (5.8.4)

Due to the symmetry of the distribution, its asymmetry is zero:

  5.8.  Uniform density law . (5.8.5)

To determine the excess, we find the fourth central point:

  5.8.  Uniform density law

from where

  5.8.  Uniform density law (5.8.6)

Determine the average arithmetic deviation:

  5.8.  Uniform density law (5.8.7)

Finally, we find the probability of hitting a random variable.   5.8.  Uniform density law distributed according to the law of uniform density on the plot   5.8.  Uniform density law representing part of a plot   5.8.  Uniform density law (fig. 5.8.4). Geometrically, this probability is the area shaded in Fig. 5.8.4. Obviously, it is equal to:

  5.8.  Uniform density law (5.8.8)

those. the ratio of the length of the segment   5.8.  Uniform density law to the entire length of the section   5.8.  Uniform density law on which uniform distribution is set.

  5.8.  Uniform density law

Fig. 5.8.4.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis