A discrete random variable x is specified by a distribution function. Theoretical material for modules "probability theory and mathematical statistics"

Discrete random Variables are random variables that take only values ​​that are distant from each other and that can be listed in advance.
Law of distribution
The distribution law of a random variable is a relationship that establishes a connection between the possible values ​​of a random variable and their corresponding probabilities.
The distribution series of a discrete random variable is the list of its possible values ​​and the corresponding probabilities.
The distribution function of a discrete random variable is the function:
,
determining for each value of the argument x the probability that the random variable X will take a value less than this x.

Expectation of a discrete random variable
,
where is the value of a discrete random variable; - the probability of a random variable accepting X values.
If a random variable takes a countable set of possible values, then:
.
Mathematical expectation of the number of occurrences of an event in n independent trials:
,

Dispersion and standard deviation of a discrete random variable
Dispersion of a discrete random variable:
or .
Variance of the number of occurrences of an event in n independent trials
,
where p is the probability of the event occurring.
Standard deviation of a discrete random variable:
.

Example 1
Draw up a law of probability distribution for a discrete random variable (DRV) X – the number of k occurrences of at least one “six” in n = 8 throws of a pair of dice. Construct a distribution polygon. Find the numerical characteristics of the distribution (distribution mode, mathematical expectation M(X), dispersion D(X), standard deviation s(X)). Solution: Let us introduce the notation: event A – “when throwing a pair of dice, a six appears at least once.” To find the probability P(A) = p of event A, it is more convenient to first find the probability P(Ā) = q of the opposite event Ā - “when throwing a pair of dice, a six never appeared.”
Since the probability of a “six” not appearing when throwing one die is 5/6, then according to the probability multiplication theorem
P(Ā) = q = = .
Respectively,
P(A) = p = 1 – P(Ā) = .
The tests in the problem follow the Bernoulli scheme, so d.s.v. magnitude X- number k the occurrence of at least one six when throwing two dice obeys the binomial law of probability distribution:

where = is the number of combinations of n By k.

The calculations carried out for this problem can be conveniently presented in the form of a table:
Probability distribution d.s.v. X º k (n = 8; p = ; q = )

Polygon (polygon) of probability distribution of a discrete random variable X shown in the figure:

Rice. Probability distribution polygon d.s.v. X=k.
The vertical line shows the mathematical expectation of the distribution M(X).

Let us find the numerical characteristics of the probability distribution of d.s.v. X. The distribution mode is 2 (here P 8(2) = 0.2932 maximum). The mathematical expectation by definition is equal to:
M(X) = = 2,4444,
Where xk = k– value taken by d.s.v. X. Variance D(X) we find the distribution using the formula:
D(X) = = 4,8097.
Standard deviation (RMS):
s( X) = = 2,1931.

Example2
Discrete random variable X given by the distribution law

Solution. If , then (third property).
If, then. Really, X can take the value 1 with probability 0.3.
If, then. Indeed, if it satisfies the inequality
, then equals the probability of an event that can occur when X will take the value 1 (the probability of this event is 0.3) or the value 4 (the probability of this event is 0.1). Since these two events are incompatible, then, according to the addition theorem, the probability of an event is equal to the sum of the probabilities 0.3 + 0.1 = 0.4. If, then. Indeed, the event is certain, therefore its probability is equal to one. So, the distribution function can be written analytically as follows:

Graph of this function:
Let us find the probabilities corresponding to these values. By condition, the probabilities of failure of the devices are equal: then the probabilities that the devices will work during the warranty period are equal:




The distribution law has the form:

LAW OF DISTRIBUTION AND CHARACTERISTICS

RANDOM VARIABLES

Random variables, their classification and methods of description.

A random quantity is a quantity that, as a result of experiment, can take on one or another value, but which one is not known in advance. For a random variable, therefore, you can only specify values, one of which it will definitely take as a result of experiment. In what follows we will call these values ​​possible values ​​of the random variable. Since a random variable quantitatively characterizes the random result of an experiment, it can be considered as a quantitative characteristic of a random event.

Random variables are usually denoted by capital letters of the Latin alphabet, for example, X..Y..Z, and their possible values ​​by corresponding small letters.

There are three types of random variables:

Discrete; Continuous; Mixed.

Discrete is a random variable whose number of possible values ​​forms a countable set. In turn, a set whose elements can be numbered is called countable. The word "discrete" comes from the Latin discretus, meaning "discontinuous, consisting of separate parts".

Example 1. A discrete random variable is the number of defective parts X in a batch of nproducts. Indeed, the possible values ​​of this random variable are a series of integers from 0 to n.

Example 2. A discrete random variable is the number of shots before the first hit on the target. Here, as in Example 1, the possible values ​​can be numbered, although in the limiting case the possible value is an infinitely large number.

Continuous is a random variable whose possible values ​​continuously fill a certain interval of the numerical axis, sometimes called the interval of existence of this random variable. Thus, on any finite interval of existence, the number of possible values ​​of a continuous random variable is infinitely large.

Example 3. A continuous random variable is the monthly electricity consumption of an enterprise.

Example 4. A continuous random variable is the error in measuring height using an altimeter. Let it be known from the operating principle of the altimeter that the error lies in the range from 0 to 2 m. Therefore, the interval of existence of this random variable is the interval from 0 to 2 m.

Law of distribution of random variables.

A random variable is considered completely specified if its possible values ​​are indicated on the numerical axis and the distribution law is established.

Law of distribution of a random variable is a relation that establishes a connection between the possible values ​​of a random variable and the corresponding probabilities.

A random variable is said to be distributed according to a given law, or subject to a given distribution law. A number of probabilities, distribution function, probability density, and characteristic function are used as distribution laws.

The distribution law gives a complete probable description of a random variable. According to the distribution law, one can judge before experiment which possible values ​​of a random variable will appear more often and which less often.

For a discrete random variable, the distribution law can be specified in the form of a table, analytically (in the form of a formula) and graphically.

The simplest form of specifying the distribution law of a discrete random variable is a table (matrix), which lists in ascending order all possible values ​​of the random variable and their corresponding probabilities, i.e.

Such a table is called a distribution series of a discrete random variable. 1

Events X 1, X 2,..., X n, consisting in the fact that as a result of the test, the random variable X will take the values ​​x 1, x 2,...x n, respectively, are inconsistent and the only possible ones (since the table lists all possible values ​​of a random variable), i.e. form a complete group. Therefore, the sum of their probabilities is equal to 1. Thus, for any discrete random variable

(This unit is somehow distributed among the values ​​of the random variable, hence the term "distribution").

The distribution series can be depicted graphically if the values ​​of the random variable are plotted along the abscissa axis, and their corresponding probabilities are plotted along the ordinate axis. The connection of the obtained points forms a broken line called a polygon or polygon of the probability distribution (Fig. 1).

Example The lottery includes: a car worth 5,000 den. units, 4 TVs costing 250 den. units, 5 video recorders worth 200 den. units A total of 1000 tickets are sold for 7 days. units Draw up a distribution law for the net winnings received by a lottery participant who bought one ticket.

Solution. Possible values ​​of the random variable X - the net winnings per ticket - are equal to 0-7 = -7 money. units (if the ticket did not win), 200-7 = 193, 250-7 = 243, 5000-7 = 4993 den. units (if the ticket has the winnings of a VCR, TV or car, respectively). Considering that out of 1000 tickets the number of non-winners is 990, and the indicated winnings are 5, 4 and 1, respectively, and using the classical definition of probability, we obtain.

X; meaning F(5); the probability that the random variable X will take values ​​from the segment . Construct a distribution polygon.

1. The distribution function F(x) of a discrete random variable is known X:

Set the law of distribution of a random variable X in the form of a table.

1. The law of distribution of a random variable is given X:

1. The probability that the store has quality certificates for the full range of products is 0.7. The commission checked the availability of certificates in four stores in the area. Draw up a distribution law, calculate the mathematical expectation and dispersion of the number of stores in which quality certificates were not found during inspection.

1. To determine the average burning time of electric lamps in a batch of 350 identical boxes, one electric lamp from each box was taken for testing. Estimate from below the probability that the average burning duration of the selected electric lamps differs from the average burning duration of the entire batch in absolute value by less than 7 hours, if it is known that the standard deviation of the burning duration of electric lamps in each box is less than 9 hours.

1. At a telephone exchange, an incorrect connection occurs with a probability of 0.002. Find the probability that among 500 connections the following will occur:

Find the distribution function of a random variable X. Construct graphs of functions and . Calculate the mathematical expectation, variance, mode and median of a random variable X.

1. An automatic machine makes rollers. It is believed that their diameter is a normally distributed random variable with a mean value of 10 mm. What is the standard deviation if, with a probability of 0.99, the diameter is in the range from 9.7 mm to 10.3 mm.

Sample A: 6 9 7 6 4 4

Sample B: 55 72 54 53 64 53 59 48

42 46 50 63 71 56 54 59

54 44 50 43 51 52 60 43

50 70 68 59 53 58 62 49

59 51 52 47 57 71 60 46

55 58 72 47 60 65 63 63

58 56 55 51 64 54 54 63

56 44 73 41 68 54 48 52

52 50 55 49 71 67 58 46

50 51 72 63 64 48 47 55

Option 17.

1. Among the 35 parts, 7 are non-standard. Find the probability that two parts taken at random will turn out to be standard.

1. Three dice are thrown. Find the probability that the sum of points on the dropped sides is a multiple of 9.

1. The word “ADVENTURE” is made up of cards, each with one letter written on it. The cards are shuffled and taken out one at a time without returning. Find the probability that the letters taken out in the order of appearance form the word: a) ADVENTURE; b) PRISONER.

1. An urn contains 6 black and 5 white balls. 5 balls are randomly drawn. Find the probability that among them there are:
1. 2 white balls;
2. less than 2 white balls;
3. at least one black ball.

1. A in one test is equal to 0.4. Find the probabilities of the following events:
1. event A appears 3 times in a series of 7 independent trials;
2. event A will appear no less than 220 and no more than 235 times in a series of 400 trials.

1. The plant sent 5,000 good-quality products to the base. The probability of damage to each product in transit is 0.002. Find the probability that no more than 3 products will be damaged during the journey.

1. The first urn contains 4 white and 9 black balls, and the second urn contains 7 white and 3 black balls. 3 balls are randomly drawn from the first urn, and 4 from the second urn. Find the probability that all the drawn balls are the same color.

1. The law of distribution of a random variable is given X:

Calculate its mathematical expectation and variance.

1. There are 10 pencils in the box. 4 pencils are drawn at random. Random value X– the number of blue pencils among those selected. Find the law of its distribution, the initial and central moments of the 2nd and 3rd orders.

1. The technical control department checks 475 products for defects. The probability that the product is defective is 0.05. Find, with probability 0.95, the boundaries within which the number of defective products among those tested will be contained.

1. At a telephone exchange, an incorrect connection occurs with a probability of 0.003. Find the probability that among 1000 connections the following will occur:
1. at least 4 incorrect connections;
2. more than two incorrect connections.

1. The random variable is specified by the distribution density function:

Find the distribution function of a random variable X. Construct graphs of functions and . Calculate the mathematical expectation, variance, mode and median of the random variable X.

1. The random variable is specified by the distribution function:

1. By sample A solve the following problems:
1. create a variation series;

· sample average;

· sample variance;

Mode and median;

Sample A: 0 0 2 2 1 4

1. calculate the numerical characteristics of the variation series:

· sample average;

· sample variance;

standard sample deviation;

· mode and median;

Sample B: 166 154 168 169 178 182 169 159

161 150 149 173 173 156 164 169

157 148 169 149 157 171 154 152

164 157 177 155 167 169 175 166

167 150 156 162 170 167 161 158

168 164 170 172 173 157 157 162

156 150 154 163 143 170 170 168

151 174 155 163 166 173 162 182

166 163 170 173 159 149 172 176

Option 18.

1. Among 10 lottery tickets, 2 are winning ones. Find the probability that out of five tickets taken at random, one will be a winner.

1. Three dice are thrown. Find the probability that the sum of the rolled points is greater than 15.

1. The word “PERIMETER” is made up of cards, each of which has one letter written on it. The cards are shuffled and taken out one at a time without returning. Find the probability that the letters taken out form the word: a) PERIMETER; b) METER.

1. An urn contains 5 black and 7 white balls. 5 balls are randomly drawn. Find the probability that among them there are:
1. 4 white balls;
2. less than 2 white balls;
3. at least one black ball.

1. Probability of an event occurring A in one trial is equal to 0.55. Find the probabilities of the following events:
1. event A will appear 3 times in a series of 5 challenges;
2. event A will appear no less than 130 and no more than 200 times in a series of 300 trials.

1. The probability of breaking the seal of a can of canned food is 0.0005. Find the probability that among 2000 cans, two will have a leak.

1. The first urn contains 4 white and 8 black balls, and the second urn contains 7 white and 4 black balls. Two balls are randomly drawn from the first urn and three balls are randomly drawn from the second urn. Find the probability that all the drawn balls are the same color.

1. Among the parts arriving for assembly, 0.1% are defective from the first machine, 0.2% from the second, 0.25% from the third, and 0.5% from the fourth. The machine productivity ratios are respectively 4:3:2:1. The part taken at random turned out to be standard. Find the probability that the part was made on the first machine.

1. The law of distribution of a random variable is given X:

Calculate its mathematical expectation and variance.

1. An electrician has three light bulbs, each of which has a defect with a probability of 0.1. The light bulbs are screwed into the socket and the current is turned on. When the current is turned on, the defective light bulb immediately burns out and is replaced by another. Find the distribution law, mathematical expectation and dispersion of the number of tested light bulbs.

1. The probability of hitting a target is 0.3 for each of 900 independent shots. Using Chebyshev's inequality, estimate the probability that the target will be hit at least 240 times and at most 300 times.

1. At a telephone exchange, an incorrect connection occurs with a probability of 0.002. Find the probability that among 800 connections the following will occur:
1. at least three incorrect connections;
2. more than four incorrect connections.

1. The random variable is specified by the distribution density function:

Find the distribution function of the random variable X. Draw graphs of the functions and . Calculate the mathematical expectation, variance, mode and median of a random variable X.

1. The random variable is specified by the distribution function:

1. By sample A solve the following problems:
1. create a variation series;
2. calculate relative and accumulated frequencies;
3. compose an empirical distribution function and plot it;
4. calculate the numerical characteristics of the variation series:

· sample average;

· sample variance;

standard sample deviation;

· mode and median;

Sample A: 4 7 6 3 3 4

1. Using sample B, solve the following problems:
1. create a grouped variation series;
2. build a histogram and frequency polygon;
3. calculate the numerical characteristics of the variation series:

· sample average;

· sample variance;

standard sample deviation;

· mode and median;

Sample B: 152 161 141 155 171 160 150 157

154 164 138 172 155 152 177 160

168 157 115 128 154 149 150 141

172 154 144 177 151 128 150 147

143 164 156 145 156 170 171 142

148 153 152 170 142 153 162 128

150 146 155 154 163 142 171 138

128 158 140 160 144 150 162 151

163 157 177 127 141 160 160 142

159 147 142 122 155 144 170 177

Option 19.

1. There are 16 women and 5 men working at the site. 3 people were selected at random using their personnel numbers. Find the probability that all selected people will be men.

2. Four coins are tossed. Find the probability that only two coins will have a “coat of arms”.

3. The word “PSYCHOLOGY” is made up of cards, each of which has one letter written on it. The cards are shuffled and taken out one at a time without returning. Find the probability that the letters taken out form a word: a) PSYCHOLOGY; b) STAFF.

4. The urn contains 6 black and 7 white balls. 5 balls are randomly drawn. Find the probability that among them there are:

a. 3 white balls;

b. less than 3 white balls;

c. at least one white ball.

5. Probability of an event occurring A in one trial is equal to 0.5. Find the probabilities of the following events:

a. event A appears 3 times in a series of 5 independent trials;

b. event A will appear at least 30 and no more than 40 times in a series of 50 trials.

6. There are 100 machines of the same power, operating independently of each other in the same mode, in which their drive is turned on for 0.8 working hours. What is the probability that at any given moment in time from 70 to 86 machines will be turned on?

7. The first urn contains 4 white and 7 black balls, and the second urn contains 8 white and 3 black balls. 4 balls are randomly drawn from the first urn, and 1 ball from the second. Find the probability that among the drawn balls there are only 4 black balls.

8. The car sales showroom receives cars of three brands daily in volumes: “Moskvich” – 40%; "Oka" - 20%; "Volga" - 40% of all imported cars. Among Moskvich cars, 0.5% have an anti-theft device, Oka – 0.01%, Volga – 0.1%. Find the probability that the car taken for inspection has an anti-theft device.

9. Numbers and are chosen at random on the segment. Find the probability that these numbers satisfy the inequalities.

10. The law of distribution of a random variable is given X:

Find the distribution function of a random variable X; meaning F(2); the probability that the random variable X will take values ​​from the interval . Construct a distribution polygon.

Examples of solving problems on the topic “Random variables”.

Task 1 . There are 100 tickets issued for the lottery. One winning of 50 USD was drawn. and ten wins of 10 USD each. Find the law of distribution of the value X - the cost of possible winnings.

Solution. Possible values ​​for X: x 1 = 0; x 2 = 10 and x 3 = 50. Since there are 89 “empty” tickets, then p 1 = 0.89, probability of winning $10. (10 tickets) – p 2 = 0.10 and to win 50 USD -p 3 = 0.01. Thus:

Easy to control: .

Task 2. The probability that the buyer has read the product advertisement in advance is 0.6 (p=0.6). Selective control of the quality of advertising is carried out by surveying buyers before the first one who has studied the advertising in advance. Draw up a distribution series for the number of buyers surveyed.

Solution. According to the problem conditions, p = 0.6. From: q=1 -p = 0.4. Substituting these values, we get: and construct a distribution series:

Task 3. A computer consists of three independently working elements: the system unit, the monitor and the keyboard. With a single sharp increase in voltage, the probability of failure of each element is 0.1. Based on the Bernoulli distribution, draw up a distribution law for the number of failed elements during a power surge in the network.

Solution. Let's consider Bernoulli distribution(or binomial): the probability that n tests, event A will appear exactly k once: , or:

IN Let's return to the task.

Possible values ​​for X (number of failures):

x 0 =0 – none of the elements failed;

x 1 =1 – failure of one element;

x 2 =2 – failure of two elements;

x 3 =3 – failure of all elements.

Since, by condition, p = 0.1, then q = 1 – p = 0.9. Using Bernoulli's formula, we get

, ,

, .

Control: .

Therefore, the required distribution law:

Problem 4. 5000 rounds produced. Probability that one cartridge is defective . What is the probability that there will be exactly 3 defective cartridges in the entire batch?

Solution. Applicable Poisson distribution: This distribution is used to determine the probability that, for very large

number of tests (mass tests), in each of which the probability of event A is very small, event A will occur k times: , Where .

Here n = 5000, p = 0.0002, k = 3. We find , then the desired probability: .

Problem 5. When firing until the first hit with hit probability p = 0.6 when firing, you need to find the probability that a hit will occur on the third shot.

Solution. Let us apply a geometric distribution: let independent trials be carried out, in each of which event A has a probability of occurrence p (and non-occurrence q = 1 – p). The test ends as soon as event A occurs.

Under such conditions, the probability that event A will occur on the kth trial is determined by the formula: . Here p = 0.6; q = 1 – 0.6 = 0.4;k = 3. Therefore, .

Problem 6. Let the distribution law of a random variable X be given:

Find the mathematical expectation.

Solution. .

Note that the probabilistic meaning of the mathematical expectation is the average value of a random variable.

Problem 7. Find the variance of the random variable X with the following distribution law:

Solution. Here .

Distribution law for the squared value of X 2 :

Required variance: .

Dispersion characterizes the measure of deviation (dispersion) of a random variable from its mathematical expectation.

Problem 8. Let a random variable be given by the distribution:

Find its numerical characteristics.

Solution: m, m 2 ,

M 2 , m.

About the random variable X we can say either: its mathematical expectation is 6.4 m with a variance of 13.04 m 2 , or – its mathematical expectation is 6.4 m with a deviation of m. The second formulation is obviously more clear.

Task 9. Random value X given by the distribution function:
.

Find the probability that as a result of the test the value X will take the value contained in the interval .

Solution. The probability that X will take a value from a given interval is equal to the increment of the integral function in this interval, i.e. . In our case and , therefore

.

Task 10. Discrete random variable X given by the distribution law:

Find the distribution function F(x ) and plot it.

Solution. Since the distribution function,

For , That

at ;

at ;

at ;

at ;

Relevant chart:

Problem 11. Continuous random variable X given by the differential distribution function: .

Find the hit probability X per interval

Solution. Note that this is a special case of the exponential distribution law.

Let's use the formula: .

Task 12. Find the numerical characteristics of a discrete random variable X specified by the distribution law:

As is known, random variable is a variable quantity 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.

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, … .

V) by using distribution functions 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 – distribution polygon (polygon) (see problem 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 discrete random variable D(X)=M2 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).

Examples of solving problems on the topic “The law of distribution of a discrete random variable”

Task 1.

1000 lottery tickets were issued: 5 of them will win 500 rubles, 10 will win 100 rubles, 20 will win 50 rubles, 50 will win 10 rubles. Determine the law of probability distribution of the random variable X - winnings per ticket.

Solution. According to the conditions of the problem, the following values ​​of the random variable X are possible: 0, 10, 50, 100 and 500.

The number of tickets without winning is 1000 – (5+10+20+50) = 915, then P(X=0) = 915/1000 = 0.915.

Similarly, we find all other probabilities: P(X=0) = 50/1000=0.05, P(X=50) = 20/1000=0.02, P(X=100) = 10/1000=0.01 , P(X=500) = 5/1000=0.005. Let us present the resulting law in the form of a table:

Let's find the mathematical expectation of the value X: M(X) = 1*1/6 + 2*1/6 + 3*1/6 + 4*1/6 + 5*1/6 + 6*1/6 = (1+ 2+3+4+5+6)/6 = 21/6 = 3.5

Task 3.

The device consists of three independently operating elements. The probability of failure of each element in one experiment is 0.1. Draw up a distribution law for the number of failed elements in one experiment, construct a distribution polygon. Find the distribution function F(x) and plot it. Find the mathematical expectation, variance and standard deviation of a discrete random variable.

Solution. 1. The discrete random variable X = (the number of failed elements in one experiment) has the following possible values: x 1 =0 (none of the device elements failed), x 2 =1 (one element failed), x 3 =2 (two elements failed ) and x 4 =3 (three elements failed).

Failures of elements are independent of each other, the probabilities of failure of each element are equal, therefore it is applicable Bernoulli's formula . Considering that, according to the condition, n=3, p=0.1, q=1-p=0.9, we determine the probabilities of the values:
P 3 (0) = C 3 0 p 0 q 3-0 = q 3 = 0.9 3 = 0.729;
P 3 (1) = C 3 1 p 1 q 3-1 = 3*0.1*0.9 2 = 0.243;
P 3 (2) = C 3 2 p 2 q 3-2 = 3*0.1 2 *0.9 = 0.027;
P 3 (3) = C 3 3 p 3 q 3-3 = p 3 =0.1 3 = 0.001;
Check: ∑p i = 0.729+0.243+0.027+0.001=1.

Thus, the desired binomial distribution law of X has the form:

We plot the possible values ​​of x i along the abscissa axis, and the corresponding probabilities p i along the ordinate axis. Let's construct points M 1 (0; 0.729), M 2 (1; 0.243), M 3 (2; 0.027), M 4 (3; 0.001). By connecting these points with straight line segments, we obtain the desired distribution polygon.

3. Let's find the distribution function F(x) = Р(Х

Graph of function F(x)

4. For binomial distribution X:
- mathematical expectation M(X) = np = 3*0.1 = 0.3;
- variance D(X) = npq = 3*0.1*0.9 = 0.27;
- standard deviation σ(X) = √D(X) = √0.27 ≈ 0.52.