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5.1. A number of distribution. Polygon distribution

Lecture



In the course section on the basic concepts of probability theory, we have already introduced into consideration the extremely important concept of a random variable. Here we will give further development of this concept and indicate the ways in which random variables can be described and characterized.

As already mentioned, a random variable is a quantity that, as a result of experience, can take on a particular value, it is not known beforehand which one. We also agreed to distinguish between random variables of discontinuous (discrete) and continuous types. Possible values ​​of discontinuous values ​​can be pre-listed. Possible values ​​of continuous quantities cannot be pre-listed and continuously fill a certain gap.

Examples of discontinuous random variables:

1) the number of occurrences of the coat of arms with three throws of a coin (possible values ​​are 0, 1, 2, 3);

2) the frequency of the emblem in the same experience (possible values   5.1.  A number of distribution.  Polygon distribution );

3) the number of failed elements in the device, consisting of five elements (possible values ​​are 0, 1, 2, 3, 4, 5);

4) the number of hits in the aircraft, sufficient to disable it (possible values ​​of 1, 2, 3, ..., n, ...);

5) the number of aircraft shot down in air combat (possible values ​​are 0, 1, 2, ..., N, where   5.1.  A number of distribution.  Polygon distribution - the total number of aircraft involved in the battle).

Examples of continuous random variables:

1) the abscissa (ordinate) of the hit point when fired;

2) the distance from the point of impact to the center of the target;

3) height meter error;

4) uptime radio tubes.

Let us further agree on random variables in large letters, and their possible values ​​in corresponding small letters. For example,   5.1.  A number of distribution.  Polygon distribution - number of hits with three shots; possible values:   5.1.  A number of distribution.  Polygon distribution .

Consider a discontinuous random variable   5.1.  A number of distribution.  Polygon distribution with possible values   5.1.  A number of distribution.  Polygon distribution . Each of these values ​​is possible, but not reliable, and the value of X can take each of them with some probability. As a result of the experiment, the value of X will take one of these values, i.e. one of the complete group of incompatible events will occur:

  5.1.  A number of distribution.  Polygon distribution (5.1.1)

Let us denote the probabilities of these events by the letters p with the corresponding indices:

  5.1.  A number of distribution.  Polygon distribution

Since incompatible events (5.1.1) form a complete group,

  5.1.  A number of distribution.  Polygon distribution ,

those. the sum of the probabilities of all possible values ​​of a random variable is equal to one. This total probability is somehow distributed between individual values. A random variable will be completely described from a probabilistic point of view if we define this distribution, i.e. We indicate exactly what probability each of the events has (5.1.1). By this we establish the so-called law of the distribution of a random variable.

The law of the distribution of a random variable is any relationship that establishes a relationship between the possible values ​​of a random variable and the corresponding probabilities. About a random variable, we will say that it is subject to this distribution law.

Set the form in which the distribution law of a discontinuous random variable can be specified.   5.1.  A number of distribution.  Polygon distribution . The simplest form of setting this law is a table that lists the possible values ​​of a random variable and the corresponding probabilities:

  5.1.  A number of distribution.  Polygon distribution

Such a table will be called a series of distribution of a random variable.   5.1.  A number of distribution.  Polygon distribution .

To give the distribution a more visual form, they often resort to its graphic representation: possible values ​​of a random variable are plotted on the abscissa axis, and probabilities of these values ​​are plotted on the ordinate axis. For clarity, the resulting points are connected by straight line segments. Such a figure is called a distribution polygon (Fig. 5.1.1). The distribution polygon, as well as the distribution series, fully characterizes the random variable; It is a form of distribution law.

  5.1.  A number of distribution.  Polygon distribution

Fig. 5.1.1.

Sometimes the so-called “mechanical” interpretation of a series of distribution is convenient. Imagine that a certain mass equal to unity is distributed along the abscissa axis so that   5.1.  A number of distribution.  Polygon distribution selected points   5.1.  A number of distribution.  Polygon distribution concentrated according to mass   5.1.  A number of distribution.  Polygon distribution . Then the distribution series is interpreted as a system of material points with some masses located on the x-axis.

Let us consider several examples of discontinuous random variables with their distribution laws.

Example 1. One experience is performed in which an event may or may not appear.   5.1.  A number of distribution.  Polygon distribution . Event probability   5.1.  A number of distribution.  Polygon distribution equal to 0.3. Considered a random variable   5.1.  A number of distribution.  Polygon distribution - the number of occurrences   5.1.  A number of distribution.  Polygon distribution in this experience (i.e., the characteristic random variable of the event   5.1.  A number of distribution.  Polygon distribution , taking the value 1 if it appears, and 0 if it does not appear). Construct the distribution series and polygon of the distribution of magnitude   5.1.  A number of distribution.  Polygon distribution .

Decision. Magnitude   5.1.  A number of distribution.  Polygon distribution It has only two values: 0 and 1. The value distribution series   5.1.  A number of distribution.  Polygon distribution has the form:

  5.1.  A number of distribution.  Polygon distribution

The distribution polygon is shown in Fig. 5.1.2.

  5.1.  A number of distribution.  Polygon distribution

Fig. 5.1.2.

Example 2. The shooter makes three shots at the target. The probability of hitting the target with each shot is 0.4. For each hit the arrow counts 5 points. Build a series of distribution of the number of points knocked out.

Decision. Denote   5.1.  A number of distribution.  Polygon distribution number of points knocked out. Possible values   5.1.  A number of distribution.  Polygon distribution :   5.1.  A number of distribution.  Polygon distribution .

The probability of these values ​​is found by the theorem on the repetition of experiments:

  5.1.  A number of distribution.  Polygon distribution

Distribution range   5.1.  A number of distribution.  Polygon distribution has the form:

  5.1.  A number of distribution.  Polygon distribution

The distribution polygon is shown in Fig. 5.1.3.

  5.1.  A number of distribution.  Polygon distribution

Fig. 5.1.3.

Example 3. The likelihood of an event   5.1.  A number of distribution.  Polygon distribution in one experience is equal   5.1.  A number of distribution.  Polygon distribution . A number of independent experiments are carried out, which continue until the first occurrence of the event.   5.1.  A number of distribution.  Polygon distribution , after which the experiments cease. Random value   5.1.  A number of distribution.  Polygon distribution - the number of experiments made. Build a series of distribution values   5.1.  A number of distribution.  Polygon distribution .

Decision. Possible values   5.1.  A number of distribution.  Polygon distribution : 1, 2, 3, ... (theoretically, they are not limited by anything). To make the value   5.1.  A number of distribution.  Polygon distribution took the value 1, it is necessary that the event   5.1.  A number of distribution.  Polygon distribution happened in the first experience; the probability of this is equal   5.1.  A number of distribution.  Polygon distribution . To make the value   5.1.  A number of distribution.  Polygon distribution took the value 2, it is necessary that in the first experience the event   5.1.  A number of distribution.  Polygon distribution did not appear, and appeared in the second; the probability of this is equal   5.1.  A number of distribution.  Polygon distribution where   5.1.  A number of distribution.  Polygon distribution , etc. Distribution range   5.1.  A number of distribution.  Polygon distribution has the form:

  5.1.  A number of distribution.  Polygon distribution

The first five ordinates of the distribution polygon for the case   5.1.  A number of distribution.  Polygon distribution shown in fig. 5.1.4.

  5.1.  A number of distribution.  Polygon distribution

Fig. 5.1.4.

Example 4. The shooter is shooting at the target before the first hit, having 4 rounds ammunition. The probability of hitting each shot is 0.6. Build a number of distribution of ammunition remaining unspent.

Decision. Random value   5.1.  A number of distribution.  Polygon distribution - The number of unused cartridges - has four possible values: 0, 1, 2 and 3. The probabilities of these values ​​are equal, respectively:

  5.1.  A number of distribution.  Polygon distribution

Distribution range   5.1.  A number of distribution.  Polygon distribution has the form:

  5.1.  A number of distribution.  Polygon distribution

The distribution polygon is shown in fig. 5.1.5.

  5.1.  A number of distribution.  Polygon distribution

Fig. 5.1.5.

Example 5. A technical device can be used in various conditions and, depending on this, requires adjustment from time to time. With a single use of the device, it may randomly fall into a favorable or unfavorable mode. In a favorable mode, the device withstands three applications without adjustment; before the fourth it has to be regulated. In an unfavorable mode, the device has to be adjusted after the first application. The probability that the device will fall into a favorable mode is 0.7, which is unfavorable, 0.3. Considered a random variable   5.1.  A number of distribution.  Polygon distribution - the number of applications of the device before adjustment. Build its distribution series.

Decision. Random value   5.1.  A number of distribution.  Polygon distribution has three possible values: 1, 2, and 3. The probability that   5.1.  A number of distribution.  Polygon distribution , is equal to the probability that at the first application the device will fall into an unfavorable mode, i.e   5.1.  A number of distribution.  Polygon distribution . To make the value   5.1.  A number of distribution.  Polygon distribution took the value 2, it is necessary that when the device is used for the first time it falls into a favorable mode, and in the second - in an unfavorable one; probability of this   5.1.  A number of distribution.  Polygon distribution . To magnitude   5.1.  A number of distribution.  Polygon distribution took the value 3, it is necessary that the first two times the device falls into a favorable mode (after the third time it will still have to be adjusted). The probability of this is   5.1.  A number of distribution.  Polygon distribution .

Distribution range   5.1.  A number of distribution.  Polygon distribution has the form:

  5.1.  A number of distribution.  Polygon distribution

The distribution polygon is shown in fig. 5.1.6.

  5.1.  A number of distribution.  Polygon distribution

Fig. 5.1.6.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis