The example discussed above allows us to conclude that the values used for analysis depend on random causes, therefore such variables are called random. In most cases, they appear as a result of observations or experiments, which are summarized in tables, in the first line of which the various observed values \u200b\u200bof the random variable X are recorded, and in the second - the corresponding frequencies. Therefore, this table is called empirical distribution of a random variable X or variational series. For the variational series, we found the mean value, variance and standard deviation.

continuous, if its values completely fill some numerical interval.

The random variable is called discrete, if all its values can be enumerated (in particular, if it takes a finite number of values).

It should be noted two characteristic properties distribution tables of a discrete random variable:

All numbers in the second row of the table are positive;

Their sum is equal to one.

In accordance with the studies carried out, it can be assumed that with an increase in the number of observations, the empirical distribution approaches the theoretical distribution given in tabular form.

An important characteristic of a discrete random variable is its mathematical expectation.

mathematical expectation discrete random variable X, taking values , , …, . with probabilities , , …, is called a number:

The mathematical expectation is also called the mean.

Other important characteristics of a random variable include variance (8) and standard deviation (9).

where: mathematical expectation of the value x.

. (9)

The graphical presentation of information is much clearer than the tabular one, so the ability of MS Excel spreadsheets to present the data placed in them in the form of various charts, graphs and histograms is used very often. So, in addition to the table, the distribution of a random variable is also depicted using distribution polygon. To do this, points with coordinates , , ... are built on the coordinate plane and connected by straight segments.

To obtain a distribution rectangle using MS Excel, you must:

1. Select the "Insert" ® "Area Chart" tab on the toolbar.

2. Activate the area for the chart that appeared on the MS Excel sheet with the right mouse button and use the “Select Data” command in the context menu.

Rice. 6. Selecting a data source

First, let's define the data range for the chart. To do this, in the appropriate area of the "Select Data Source" dialog box, enter the range C6:I6 (it contains the frequency values called Row1, Fig. 7).

Rice. 7. Add row 1

To change the name of a series, select the button to change the "Legend elements (series)" area (see Fig. 7) and name it .

In order to add a label for the X axis, use the "Edit" button in the "Horizontal axis labels (categories)" area
(Fig. 8) and indicate the values of the series (range $C$6:$I$6).

Rice. 8. The final view of the dialog box "Select data source"

Selecting a button in the Select Data Source dialog box
(Fig. 8) will allow you to obtain the required polygon of the distribution of a random variable (Fig. 9).

Rice. 9. Polygon distribution of a random variable

Let's make some changes to the design of the received graphic information:

Add an x-axis label;

Edit the label of the Y axis;

- Let's add a title for the "Distribution Polygon" chart.

To do this, select the “Work with charts” tab in the toolbar area, the “Layout” tab and in the toolbar that appears, the corresponding buttons: “Chart name”, “Axis names” (Fig. 10).

Rice. 10. The final form of the polygon of the distribution of a random variable

Random variable A quantity is called which, as a result of an experiment, can take on one or another value that is not known in advance. Random variables are discontinuous (discrete) And continuous type. Possible values of discontinuous quantities can be enumerated in advance. The possible values of continuous quantities cannot be enumerated in advance and continuously fill a certain gap.

An example of discrete random variables:

1) The number of appearance of the coat of arms in three coin tosses. (possible values are 0;1;2;3)

2) The frequency of the appearance of the coat of arms in the same experiment. (possible values)

3) The number of failed elements in a device consisting of five elements. (Possible values are 0;1;2;3;4;5)

Examples of continuous random variables:

1) Abscissa (ordinate) of the point of impact when fired.

2) The distance from the point of impact to the center of the target.

3) Time of non-failure operation of the device (radio tubes).

Random variables are denoted by capital letters, and their possible values by the corresponding small letters. For example, X is the number of hits with three shots; possible values: X 1 =0, X 2 =1, X 3 =2, X 4 =3.

Consider a discontinuous random variable X with possible values X 1 , X 2 , … , X n . Each of these values is possible, but not certain, and the value of X can take each of them with some probability. As a result of the experiment, the quantity X will take one of these values, that is, one of the complete group of incompatible events will occur.

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

Since incompatible events form a complete group, then

that is, the sum of the probabilities of all possible values of the random variable is equal to 1. This total probability is somehow distributed among the individual values. A random variable will be completely described from a probabilistic point of view if we specify this distribution, that is, we indicate exactly what probability each of the events has. (This will establish the so-called law of distribution of random variables.)

The law of distribution of a random variable Any relation that establishes a connection between the possible values of a random variable and the corresponding probability is called. (About a random variable, we will say that it is subject to a given distribution law)

The simplest form of specifying the law of distribution of a random variable is a table that lists the possible values of a random variable and their corresponding probabilities.

Table 1.

random variables. Distribution polygon

Random variables: discrete and continuous.

When conducting a stochastic experiment, a space of elementary events is formed - the possible outcomes of this experiment. It is considered that on this space of elementary events random value X, if a law (rule) is given according to which a number is assigned to each elementary event. Thus, the random variable X can be considered as a function defined on the space of elementary events.

■ Random- a value that, during each test, takes on one or another numerical value (it is not known in advance which one), depending on random causes that cannot be taken into account in advance. Random variables are denoted by capital letters of the Latin alphabet, and possible values of a random variable are denoted by small letters. So, when a dice is thrown, an event occurs associated with the number x, where x is the number of points rolled. The number of points is a random value, and the numbers 1, 2, 3, 4, 5, 6 are the possible values of this value. The distance that a projectile will fly when fired from a gun is also a random variable (it depends on the installation of the sight, the strength and direction of the wind, temperature, and other factors), and the possible values of this quantity belong to a certain interval (a; b).

■ Discrete random variable- a random variable that takes on separate, isolated possible values with certain probabilities. The number of possible values of a discrete random variable can be finite or infinite.

■ Continuous random variable is a random variable that can take on all values from some finite or infinite interval. The number of possible values of a continuous random variable is infinite.

For example, the number of points dropped when throwing a dice, the score for a control work are discrete random variables; the distance that a projectile flies when firing from a gun, the measurement error of the indicator of the time of assimilation of educational material, the height and weight of a person are continuous random variables.

Distribution law of a random variable– correspondence between the possible values of a random variable and their probabilities, i.e. each possible value x i is associated with the probability p i with which the random variable can take this value. The law of distribution of a random variable can be given tabularly (in the form of a table), analytically (in the form of a formula) and graphically.

Let a discrete random variable X take the values x 1 , x 2 , …, x n with probabilities p 1 , p 2 , …, p n respectively, i.e. P(X=x 1) = p 1 , P(X=x 2) = p 2 , …, P(X=x n) = p n . With a tabular assignment of the distribution law of this value, the first row of the table contains the possible values x 1, x 2, ..., x n, and the second - their probabilities

X	x 1	x2	…	x n
p	p1	p2	…	p n

As a result of the test, the discrete random variable X takes one and only one of the possible values, so the events X=x 1 , X=x 2 , …, X=x n form a complete group of pairwise incompatible events, and, therefore, the sum of the probabilities of these events is equal to one , i.e. p 1 + p 2 + ... + p n \u003d 1.

The law of distribution of a discrete random variable. Polygon (polygon) distribution.

As you know, a random variable is a variable that can take on certain values depending on the case. Random variables are denoted by capital letters of the Latin alphabet (X, Y, Z), and their values - by the corresponding lowercase letters (x, y, z). Random variables are divided into discontinuous (discrete) and continuous.

A discrete random variable is a random variable that takes only a finite or infinite (countable) set of values with certain non-zero probabilities.

The distribution law of a discrete random variable is a function that connects the values of a random variable with their corresponding probabilities. The distribution law can be specified in one of the following ways.

1. The distribution law can be given by the table:

where λ>0, k = 0, 1, 2, … .

c) using the distribution function F(x), which determines for each value x the probability that the random variable X will take on a value less than x, i.e. F(x) = P(X< x).

Properties of the function F(x)

3. The distribution law can be specified graphically - by a distribution polygon (polygon) (see task 3).

Note that in order to solve some problems, it is not necessary to know the distribution law. In some cases, it is enough to know one or more numbers that reflect the most important features of the distribution law. It can be a number that has the meaning of the "average value" of a random variable, or a number that shows the average size of the deviation of a random variable from its average value. Numbers of this kind are called numerical characteristics of a random variable.

The main numerical characteristics of a discrete random variable:

Mathematical expectation (average value) of a discrete random variable M(X)=Σ x i p i .
For binomial distribution M(X)=np, for Poisson distribution M(X)=λ
Dispersion of a discrete random variable D(X)= M 2 or D(X) = M(X 2)− 2 . The difference X–M(X) is called the deviation of a random variable from its mathematical expectation.
For binomial distribution D(X)=npq, for Poisson distribution D(X)=λ
Standard deviation (standard deviation) σ(X)=√D(X).

· For clarity of representation of the variation series, its graphic representations are of great importance. Graphically, a variational series can be displayed as a polygon, a histogram, and a cumulate.

· A distribution polygon (literally, a distribution polygon) is called a broken line, which is built in a rectangular coordinate system. The value of the feature is plotted on the abscissa, the corresponding frequencies (or relative frequencies) - along the ordinate. Points (or ) are connected by line segments and a distribution polygon is obtained. Most often, polygons are used to display discrete variation series, but they can also be used for interval series. In this case, points corresponding to the midpoints of these intervals are plotted on the abscissa axis.

X i	x1	x2	…	X n
Pi	P1	P2	…	P n

Such a table is called near distribution random variables.

To give the distribution series a more visual form, they resort to its graphical representation: the possible values of a random variable are plotted along the abscissa axis, and the probabilities of these values are plotted along the ordinate axis. (For clarity, the obtained points are connected by line segments.)

Figure 1 - distribution polygon

Such a figure is called distribution polygon. The distribution polygon, like the distribution series, completely characterizes the random variable; it is a form of the law of distribution.

Example:

one experiment is performed in which event A may or may not appear. Probability of event A = 0.3. A random variable X is considered - the number of occurrences of event A in this experiment. It is necessary to build a series and a polygon of the distribution of X.

Table 2.

X i
Pi	0,7	0,3

Figure 2 - Distribution function

distribution function is a universal characteristic of a random variable. It exists for all random variables: both discontinuous and non-discontinuous. The distribution function completely characterizes a random variable from a probabilistic point of view, that is, it is one of the forms of the distribution law.

To quantify this probability distribution, it is convenient to use not the probability of the event X=x, but the probability of the event X

The distribution function F(x) is sometimes also called the integral distribution function or the integral distribution law.

Properties of the distribution function of a random variable

1. The distribution function F(x) is a non-decreasing function of its argument, that is, for ;

2. At minus infinity:

3. On plus infinity:

Figure 3 - graph of the distribution function

Distribution function plot in the general case, it is a graph of a non-decreasing function, the values of which start from 0 and reach 1.

Knowing the distribution series of a random variable, it is possible to construct the distribution function of a random variable.

Example:

for the conditions of the previous example, construct a distribution function of a random variable.

Let's construct the distribution function X:

Figure 4 - distribution function X

distribution function of any discontinuous discrete random variable there is always a discontinuous step function whose jumps occur at points corresponding to the possible values of the random variable and are equal to the probabilities of these values. The sum of all jumps in the distribution function is 1.

As the number of possible values of the random variable increases and the intervals between them decrease, the number of jumps becomes larger, and the jumps themselves become smaller:

Figure 5

The step curve becomes smoother:

Figure 6

A random variable gradually approaches a continuous value, and its distribution function approaches a continuous function. There are also random variables whose possible values continuously fill a certain gap, but for which the distribution function is not everywhere continuous. And at some points it breaks. Such random variables are called mixed.

Figure 7

Task 14. In the cash lottery, 1 win of 1,000,000 rubles is played, 10 wins of 100,000 rubles each. and 100 winnings of 1000 rubles. with a total number of tickets 10000. Find the law of distribution of random winnings X for the owner of one lottery ticket.

Solution. Possible values for X: X 1 = 0; X 2 = 1000; X 3 = 100000;

X 4 \u003d 1000000. Their probabilities are respectively equal: R 2 = 0,01; R 3 = 0,001; R 4 = 0,0001; R 1 = 1 – 0,01 – 0,001 – 0,0001 = 0,9889.

Therefore, the distribution law of the payoff X can be given by the following table:

Task 15. Discrete random variable X given by the distribution law:

Construct a distribution polygon.

Solution. We construct a rectangular coordinate system, and along the abscissa axis we will plot the possible values x i, and along the y-axis - the corresponding probabilities p i. Let's build points M 1 (1;0,2), M 2 (3;0,1), M 3 (6; 0.4) and M 4 (8; 0.3). Connecting these points with line segments, we obtain the desired distribution polygon.

§2. Numerical characteristics of random variables

A random variable is completely characterized by its distribution law. An average description of a random variable can be obtained using its numerical characteristics

2.1. Expected value. Dispersion.

Let a random variable take on values with probabilities respectively .

Definition. The mathematical expectation of a discrete random variable is the sum of the products of all its possible values and the corresponding probabilities:

Properties of mathematical expectation.

The dispersion of a random variable around the mean value is characterized by the variance and standard deviation.

The dispersion of a random variable is the mathematical expectation of the squared deviation of a random variable from its mathematical expectation:

For calculations, the following formula is used

Dispersion properties.

2. , where are mutually independent random variables.

3. Standard deviation.

Task 16. Find the mathematical expectation of a random variable Z = X+ 2Y, if the mathematical expectations of random variables are known X And Y: M(X) = 5, M(Y) = 3.

Solution. We use the properties of mathematical expectation. Then we get:

M(X+ 2Y)= M(X) + M(2Y) = M(X) + 2M(Y) = 5 + 2 . 3 = 11.

Task 17. Variance of a random variable X equal to 3. Find the variance of random variables: a) –3 X; b) 4 X + 3.

Solution. Let's apply properties 3, 4 and 2 of dispersion. We have:

A) D(–3X) = (–3) 2 D(X) = 9D(X) = 9 . 3 = 27;

b) D(4X + 3) = D(4X) + D(3) = 16D(X) + 0 = 16 . 3 = 48.

Task 18. Given an independent random variable Y is the number of points scored by throwing a die. Find the distribution law, mathematical expectation, variance and standard deviation of a random variable Y.

Solution. Random variable distribution table Y looks like:

Then M(Y) = 1 1/6 + 2 1/6 + 3 1/6+ 4 1/6+ 5 1/6+ 6 1/6 = 3.5;

D(Y) \u003d (1 - 3.5) 2 1/6 + (2 - 3.5) 2 / 6 + (3 - 3.5) 2 1/6 + (4 - 3.5) 2 / 6 + (5 - -3.5) 2 1/6 + (6 - 3.5) 2. 1/6 \u003d 2.917; σ (Y) =Ö 2,917 = 1,708.