Series and polygon of distribution of a discrete random variable. Great encyclopedia of oil and gas

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

As already mentioned, a random variable is a quantity that, as a result of experiment, can take on one or another value, but it is not known in advance which one. We also agreed to distinguish between random variables of discontinuous (discrete) and continuous types. Possible values of discontinuous quantities can be listed in advance. Possible values of continuous quantities cannot be listed in advance and continuously fill a certain gap.

Examples of discontinuous random variables:

1) the number of appearances of the coat of arms during three coin tosses (possible values 0, 1, 2, 3);

2) frequency of 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);

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

5) the number of aircraft shot down in air combat (possible values 0, 1, 2, ..., N, where is the total number of aircraft participating in the battle).

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) height meter error;

4) failure-free operation time of the radio tube.

Let us agree in what follows to denote random variables by capital letters, and their possible values by corresponding small letters. For example, – the number of hits with three shots; possible values: .

Let us consider a discontinuous random variable with possible values . Each of these values is possible, but not certain, and the value X can take each of them with some probability. As a result of the experiment, the value X will take one of these values, i.e. 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 (5.1.1) form a complete group, then

those. the sum of the probabilities of all possible values of a random variable is equal to one. This total probability is somehow distributed among the individual values. The random variable will be fully described from a probabilistic point of view if we specify this distribution, i.e. Let us indicate exactly what probability each of the events (5.1.1) has. With this we will establish the so-called law of distribution of a random variable.

The law of distribution of a random variable is any relationship that establishes a connection between the possible values of a random variable and the corresponding probabilities. We will say about a random variable that it is subject to a given distribution law.

Let us establish the form in which the distribution law of a discontinuous random variable can be specified. The simplest form of specifying this law is a table that lists the possible values of a random variable and their corresponding probabilities:

We will call such a table a distribution series of a random variable.

Sometimes the so-called “mechanical” interpretation of the distribution series is convenient. Let us imagine that a certain mass equal to one is distributed along the abscissa axis in such a way that the masses are concentrated at individual points, respectively. Then the distribution series is interpreted as a system of material points with some masses located on the abscissa axis.

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

Example 1. One experiment is performed in which the event may or may not appear. The probability of the event is 0.3. A random variable is considered - the number of occurrences of an event in a given experiment (i.e. a characteristic random variable of an event, taking the value 1 if it appears, and 0 if it does not appear). Construct a distribution series and a magnitude distribution polygon.

Solution. The quantity has only two values: 0 and 1. The distribution series of the quantity has the form:

The distribution polygon is shown in Fig. 5.1.2.

Example 2. A shooter fires three shots at a target. The probability of hitting the target with each shot is 0.4. For each hit the shooter gets 5 points. Construct a distribution series for the number of points scored.

Solution. Let us denote the number of points scored. Possible values: .

We find the probability of these values using the theorem on repetition of experiments:

The value distribution series has the form:

The distribution polygon is shown in Fig. 5.1.3.

Example 3. The probability of an event occurring in one experiment is equal to . A series of independent experiments are carried out, which continue until the first occurrence of the event, after which the experiments are stopped. Random variable – the number of experiments performed. Construct a series of distribution of the value.

Solution. Possible values: 1, 2, 3, ... (theoretically they are not limited by anything). In order for a quantity to take on the value 1, it is necessary that the event occur in the first experiment; the probability of this is equal. In order for a quantity to take on the value 2, it is necessary that the event does not appear in the first experiment, but does appear in the second; the probability of this is equal to , where , etc. The value distribution series has the form:

The first five ordinates of the distribution polygon for the case are shown in Fig. 5.1.4.

Example 4. A shooter shoots at a target until the first hit, having 4 rounds of ammunition. The probability of hitting each shot is 0.6. Construct a distribution series for the amount of ammunition remaining unspent.

Random variables: discrete and continuous.

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

■ Random variable- a quantity that, during each test, takes on one or another numerical value (it is not known in advance which one), depending on random reasons 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 throwing a die, an event occurs associated with the number x, where x is the number of points rolled. The number of points is a random variable, and the numbers 1, 2, 3, 4, 5, 6 are possible values of this value. The distance that a projectile will travel when fired from a gun is also a random variable (depending on the installation of the sight, the strength and direction of the wind, temperature and other factors), and the possible values of this value 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– a random variable that can take 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 rolled when throwing a dice, the score for a test are discrete random variables; the distance that a projectile flies when firing from a gun, the measurement error of the indicator of time to master educational material, the height and weight of a person are continuous random variables.

Distribution law of a random variable– correspondence between 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 distribution law of a random variable can be specified tabularly (in the form of a table), analytically (in the form of a formula), and graphically.

Let a discrete random variable X take 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. When specifying the distribution law of this quantity in a table, the first row of the table contains possible values x 1 , x 2 , ..., x n , and the second row contains their probabilities

X	x 1	x 2	…	x n
p	p 1	p2	…	p n

As a result of the test, a discrete random variable X takes on one and only one of the possible values, therefore 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 =1.

Distribution law of a discrete random variable. Distribution polygon (polygon).

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 are denoted by 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.

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 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 to solve some problems it is not necessary to know the distribution law. In some cases, it is enough to know one or several numbers that reflect the most important features of the distribution law. This can be a number that has the meaning of the “average value” of a random variable, or a number showing the average size of the deviation of a random variable from its mean value. Numbers of this kind are called numerical characteristics of a random variable.

Basic 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
Mean square deviation (standard deviation) σ(X)=√D(X).

· For clarity of presentation of a variation series, its graphic images are of great importance. Graphically, a variation series can be depicted as a polygon, histogram and cumulate.

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

The example discussed above allows us to conclude that the values used for analysis depend on random reasons, therefore such variables are called random. In most cases, they arise as a result of observations or experiments that are tabulated, in the first row of which the various observed values of the random variable X are recorded, and in the second the corresponding frequencies. That's why this table is called empirical distribution of random variable X or variation series. For the variation series we found the mean, dispersion and standard deviation.

continuous, if its values completely fill a certain numerical interval.

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

There are two things to note characteristic properties discrete random variable distribution tables:

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

Their sum is equal to one.

In accordance with the conducted research, it can be assumed that with an increase in the number of observations, the empirical distribution approaches the theoretical one, 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 the number:

The expected value 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)

A graphical representation of information is much more visual than a tabular one, so the ability of MS Excel spreadsheets to present the data contained 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 constructed 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 chart area that appears 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, enter the range C6:I6 into the appropriate area of the “Select Data Source” dialog box (it presents the frequency values called Series1, Fig. 7).

Rice. 7. Adding row 1

To change the name of a series, you must select the button change the area “Legend elements (series)” (see Fig. 7) and name it.

In order to add an X-axis label, you must 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. Final view of the “Select data source” dialog box

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

Rice. 9. Distribution polygon of a random variable

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

Let's add a label for the X axis;

Let's edit the Y axis label;

- Let's add a title for the diagram “Distribution polygon”.

To do this, select the “Working with Charts” tab in the toolbar area, the “Layout” tab and in the toolbar that appears, the corresponding buttons: “Chart title”, “Axes titles” (Fig. 10).

Rice. 10. Final view of the random variable distribution polygon

Answer: Consider a discontinuous random variable X with possible values. Each of these values is possible, but not certain, and the value X can accept each of them with some probability. As a result of the experiment, the value X will take one of these values, i.e. one of the complete group of incompatible events will occur:

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

That is, the probability distribution of various values can be specified by a distribution table, in which all the values taken by a given discrete random variable are indicated in the top line, and the probabilities of the corresponding values are indicated in the bottom line. Since incompatible events (3.1) form a complete group, then, i.e., the sum of the probabilities of all possible values of the random variable is equal to one. The probability distribution of continuous random variables cannot be presented in the form of a table, since the number of values of such random variables is infinite even in a limited interval. Moreover, the probability of getting any particular value is zero. A random variable will be fully described from a probabilistic point of view if we specify this distribution, that is, we indicate exactly what probability each of the events has. With this we will establish the so-called law of distribution of a random variable. The law of distribution of a random variable is any relationship that establishes a connection between the possible values of a random variable and the corresponding probabilities. We will say about a random variable that it is subject to a given distribution law. Let us establish the form in which the distribution law of a discontinuous random variable can be specified X. The simplest form of specifying this law is a table that lists the possible values of a random variable and their corresponding probabilities:

x i	x 1	x 2	× × ×	x n
p i	p 1	p 2	× × ×	p n

We will call such a table a series of distributions of a random variable X.

Rice. 3.1

To give the distribution series a more visual appearance, they often resort to its graphical representation: the possible values of the random variable are plotted along the abscissa axis, and the probabilities of these values are plotted along the ordinate axis. For clarity, the resulting points are connected by straight segments. Such a figure is called a distribution polygon (Fig. 3.1). The distribution polygon, as well as the distribution series, completely characterizes the random variable. it is one of the forms of the law of distribution. Sometimes the so-called “mechanical” interpretation of the distribution series is convenient. Let us imagine that a certain mass equal to unity is distributed along the abscissa axis so that in n masses are concentrated at individual points, respectively . Then the distribution series is interpreted as a system of material points with some masses located on the abscissa axis.

Experience is any implementation of certain conditions and actions under which the random phenomenon being studied is observed. Experiments can be characterized qualitatively and quantitatively. A random quantity is a quantity that, as a result of experiment, can take on one or another value, and it is not known in advance which one.

Random variables are usually denoted (X,Y,Z), and the corresponding values (x,y,z)

Discrete are random variables that take individual values isolated from each other that can be overestimated. Continuous quantities whose possible values continuously fill a certain range. The law of distribution of a random variable is any relation that establishes a connection between the possible values of random variables and the corresponding probabilities. Distribution row and polygon. The simplest form of the distribution law of a discrete quantity is a distribution series. The graphical interpretation of the distribution series is the distribution polygon.

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Series and polygon of distribution of a discrete random variable. Great encyclopedia of oil and gas

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