The law of distribution of the random variable x is given. Discrete random variable and its distribution function

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 = )

k

Pn(k)

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

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

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:

Definition 2.3. A random variable, denoted by X, is called discrete if it takes on a finite or countable set of values, i.e. set – a finite or countable set.

Let's consider examples of discrete random variables.

1. Two coins are tossed once. The number of emblems in this experiment is a random variable X. Its possible values are 0,1,2, i.e. – a finite set.

2. The number of ambulance calls within a given period of time is recorded. Random variable X– number of calls. Its possible values are 0, 1, 2, 3, ..., i.e. =(0,1,2,3,...) is a countable set.

3. There are 25 students in the group. On a certain day, the number of students who came to class is recorded - a random variable X. Its possible values: 0, 1, 2, 3, ...,25 i.e. =(0, 1, 2, 3, ..., 25).

Although all 25 people in example 3 cannot miss classes, the random variable X can take this value. This means that the values of a random variable have different probabilities.

Let's consider a mathematical model of a discrete random variable.

Let a random experiment be carried out, which corresponds to a finite or countable space of elementary events. Let us consider the mapping of this space onto the set of real numbers, i.e., let us assign to each elementary event a certain real number , . The set of numbers can be finite or countable, i.e. or

A system of subsets, which includes any subset, including a one-point one, forms an -algebra of a numerical set ( – finite or countable).

Since any elementary event is associated with certain probabilities p i(in the case of finite everything), and , then each value of a random variable can be associated with a certain probability p i, such that .

Let X is an arbitrary real number. Let's denote R X (x) the probability that the random variable X took a value equal to X, i.e. P X (x)=P(X=x). Then the function R X (x) can take positive values only for those values X, which belong to a finite or countable set , and for all other values the probability of this value P X (x) = 0.

So, we have defined the set of values, -algebra as a system of any subsets and for each event ( X = x) compared the probability for any, i.e. constructed a probability space.

For example, the space of elementary events of an experiment consisting of tossing a symmetrical coin twice consists of four elementary events: , where

When the coin was tossed twice, two tails appeared; when the coin was tossed twice, two coats of arms fell;

On the first toss of the coin, a hash came up, and on the second, a coat of arms;

On the first toss of the coin, the coat of arms came up, and on the second, the hash mark.

Let the random variable X– number of grating dropouts. It is defined on and the set of its values . All possible subsets, including single-point ones, form an algebra, i.e. =(Ø, (1), (2), (0,1), (0,2), (1,2), (0,1,2)).

Probability of an event ( X=x i}, і = 1,2,3, we define as the probability of the occurrence of an event that is its prototype:

Thus, on elementary events ( X = x i) set a numerical function R X, So .

Definition 2.4. The distribution law of a discrete random variable is a set of pairs of numbers (x i, р i), where x i are the possible values of the random variable, and р i are the probabilities with which it takes these values, and .

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

			…		…
			…		…

Such a table is called a distribution series. To give the distribution series a more visual appearance, it is depicted graphically: on the axis Oh dots x i and draw perpendiculars of length from them p i. The resulting points are connected and a polygon is obtained, which is one of the forms of the distribution law (Fig. 2.1).

Thus, to specify a discrete random variable, you need to specify its values and the corresponding probabilities.

Example 2.2. The cash slot of the machine is triggered each time a coin is inserted with the probability r. Once it is triggered, the coins do not come down. Let X– the number of coins that must be inserted before the machine’s cash slot is triggered. Construct a series of distribution of a discrete random variable X.

Solution. Possible values of a random variable X: x 1 = 1, x 2 = 2,..., x k = k, ... Let's find the probabilities of these values: p 1– the probability that the money receiver will operate the first time it is lowered, and p 1 = p; p 2 – the probability that two attempts will be made. To do this, it is necessary that: 1) the money receiver does not work on the first attempt; 2) on the second try it worked. The probability of this event is (1–р)р. Likewise and so on, . Distribution range X will take the form

	1	2	3	…	To	…
	r	qp	q 2 p		q r -1 p

Note that the probabilities r k form a geometric progression with the denominator: 1–p=q, q<1, therefore this probability distribution is called geometric.

Let us further assume that a mathematical model has been constructed experiment described by a discrete random variable X, and consider calculating the probabilities of the occurrence of arbitrary events.

Let an arbitrary event contain a finite or countable set of values x i: A= {x 1, x 2,..., x i, ...) .Event A can be represented as a union of incompatible events of the form: . Then, using Kolmogorov’s axiom 3 , we get

since we determined the probabilities of the occurrence of events to be equal to the probabilities of the occurrence of events that are their prototypes. This means that the probability of any event , , can be calculated using the formula, since this event can be represented in the form of a union of events, where .

Then the distribution function F(x) = Р(–<Х<х) is found by the formula. It follows that the distribution function of a discrete random variable X is discontinuous and increases in jumps, i.e. it is a step function (Fig. 2.2):

If the set is finite, then the number of terms in the formula is finite, but if it is countable, then the number of terms is countable.

Example 2.3. The technical device consists of two elements that operate independently of each other. The probability of failure of the first element during time T is 0.2, and the probability of failure of the second element is 0.1. Random variable X– the number of failed elements during time T. Find the distribution function of the random variable and plot its graph.

Solution. The space of elementary events of an experiment consisting of studying the reliability of two elements of a technical device is determined by four elementary events , , , : – both elements are operational; – the first element is working, the second is faulty; – the first element is faulty, the second is working; – both elements are faulty. Each of the elementary events can be expressed through elementary events of spaces And , where – the first element is operational; – the first element has failed; – the second element is working; – the second element has failed. Then, and since the elements of a technical device work independently of each other, then

8. What is the probability that the values of a discrete random variable belong to the interval ?

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

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.

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

X	–28	–20	–12	–4
p	0,22	0,44	0,17	0,1	0,07

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.

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.

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.

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.

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

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

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.

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

2 white balls;
less than 2 white balls;
at least one black ball.

A in one test is equal to 0.4. Find the probabilities of the following events:

event A appears 3 times in a series of 7 independent trials;
event A will appear no less than 220 and no more than 235 times in a series of 400 trials.

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.

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.

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

Calculate its mathematical expectation and variance.

There are 10 pencils in a box. 4 pencils are drawn at random. Random variable 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.

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.

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:

at least 4 incorrect connections;
more than two incorrect connections.

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.

The random variable is specified by the distribution function:

By sample A solve the following problems:

create a variation series;

· sample average;

· sample variance;

Mode and median;

Sample A: 0 0 2 2 1 4

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.

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

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

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.

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

4 white balls;
less than 2 white balls;
at least one black ball.

Probability of an event occurring A in one trial is equal to 0.55. Find the probabilities of the following events:

event A will appear 3 times in a series of 5 challenges;
event A will appear no less than 130 and no more than 200 times in a series of 300 trials.

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

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.

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.

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

Calculate its mathematical expectation and variance.

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.

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.

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:

at least three incorrect connections;
more than four incorrect connections.

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.

The random variable is specified by the distribution function:

By sample A solve the following problems:

create a variation series;
calculate relative and accumulated frequencies;
compose an empirical distribution function and plot it;
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

Using sample B, solve the following problems:

create a grouped variation series;
build a histogram and frequency polygon;
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:

X
p	0,1	0,2	0,3	0,4

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.

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.

On this page we have collected examples of educational solutions problems about discrete random variables. This is a fairly extensive section: various distribution laws (binomial, geometric, hypergeometric, Poisson and others), properties and numerical characteristics are studied; for each distribution series, graphical representations can be built: polygon (polygon) of probabilities, distribution function.

Below you will find examples of decisions about discrete random variables, in which you need to apply knowledge from previous sections of probability theory to draw up a distribution law, and then calculate the mathematical expectation, variance, standard deviation, construct a distribution function, answer questions about the DSV, etc. p.

Examples for popular probability distribution laws:

Calculators for DSV characteristics

Calculation of mathematical expectation, dispersion and standard deviation of DSV.

Solved problems about DSV

Distributions close to geometric

Task 1. There are 4 traffic lights along the path of the car, each of which prohibits further movement of the car with a probability of 0.5. Find the distribution series of the number of traffic lights passed by the car before the first stop. What are the mathematical expectation and variance of this random variable?

Task 2. The hunter shoots at the game until the first hit, but manages to fire no more than four shots. Draw up a distribution law for the number of misses if the probability of hitting the target with one shot is 0.7. Find the variance of this random variable.

Task 3. The shooter, having 3 cartridges, shoots at the target until the first hit. The hit probabilities for the first, second and third shots are 0.6, 0.5, 0.4, respectively. S.V. $\xi$ - number of remaining cartridges. Compile a distribution series of a random variable, find the mathematical expectation, variance, standard deviation of the random variable, construct the distribution function of the random variable, find $P(|\xi-m| \le \sigma$.

Task 4. The box contains 7 standard and 3 defective parts. They take out the parts sequentially until the standard one appears, without returning them back. $\xi$ is the number of defective parts retrieved.
Draw up a distribution law for a discrete random variable $\xi$, calculate its mathematical expectation, variance, standard deviation, draw a distribution polygon and a graph of the distribution function.

Tasks with independent events

Task 5. 3 students appeared for the re-exam in probability theory. The probability that the first person will pass the exam is 0.8, the second - 0.7, and the third - 0.9. Find the distribution series of the random variable $\xi$ of the number of students who passed the exam, plot the distribution function, find $M(\xi), D(\xi)$.

Task 6. The probability of hitting the target with one shot is 0.8 and decreases with each shot by 0.1. Draw up a distribution law for the number of hits on a target if three shots are fired. Find the expected value, variance and S.K.O. this random variable. Draw a graph of the distribution function.

Task 7. 4 shots are fired at the target. The probability of a hit increases as follows: 0.2, 0.4, 0.6, 0.7. Find the law of distribution of the random variable $X$ - the number of hits. Find the probability that $X \ge 1$.

Task 8. Two symmetrical coins are tossed and the number of coats of arms on both top sides of the coins is counted. We consider a discrete random variable $X$ - the number of coats of arms on both coins. Write down the distribution law of the random variable $X$, find its mathematical expectation.

The law of distribution of the random variable x is given. Discrete random variable and its distribution function

Calculators for DSV characteristics

Solved problems about DSV

Distributions close to geometric

Tasks with independent events

Other problems and laws of distribution of DSV