Physical model of a binamic distribution. The binomial law was subdivided. Now I need

Let's take a look at the design of Bernoulli's scheme, tobto. a series of repeated independent samplings is carried out, in the skin of those it is given the same test. І for skin testing, there are only two results:

1) subya A - success;

2) podia - unsuccessful,

with constant emotions

Let us introduce a discrete vipad value X - “the number of appearances of sub A at P testing” and we know the law of rozpodіlu tsієї vipadkovy size. The X value can take on a value

Ymovirnist that the vipadka value X is the value of x k know the Bernoulli formula

The law of subdivision of a discrete variable value, which is determined by the Bernoulli formula (1), is called bіnomnym law rozpodіlu. Postiyni P і R (q=1-p), which enter formula (1) are called the parameters of the binamic subdivision.

The name "binomial rozpodil" is due to the fact that the right part of equality (1) is the main member of the expansion of Newton's binomial, tobto.

(2)

And the shards p+q=1, then the rights of parity (2) are equal 1

Tse means what

(4)

Equity (3) has the first term q n in the right part means the immovability of what is in P viprobuvannyah podіya A not z'appear once again, another member imovirnіst that podіya A z'appear once, the third member - imovіrnіst, scho podіya A z'appear two and find the remaining member r p- imovіrnіst that podіya A z'appear exactly P razіv.

The binomial law is given under the discrete variable value at the table:

X	0	1	…	k	…	n
R	q n		…		…	r p

The main numerical characteristics of the binomial distribution:

1) mathematical refinement (5)

2) dispersion (6)

3) mean square deviation (7)

4) the most recent number of appearances k 0- tse number yakomu for a given P vіdpovidaє maximum bіnomna ymovirnіst

When tasked Pі R the whole number is marked by irregularities

(8)

what number pr+r not є qіlim, then k 0 the second part of the number pr+r- whole number, then k 0 may have two values

The binomial law of variability of stagnation is found in the theory of shooting, the theory and practice of statistical control over the quality of production, the theory of mass service, the theory of surplus, and so on. Whose law can stagnate in all types, if there is a sequence of independent trials.

Example 1: By re-verification of quality, it was established that out of 100 leather fittings, there were no defects, 90 pieces in the average. Add up the binomial law by subdividing the number of imitations of the number of auxiliaries from the commodities 4.

Solution: Podiya A - the appearance of some kind of perverted tse - “pridbany navmannya prilad yakіsny”. For the mental task of the main parameters of the bіnomic distribution:

Vipadkov value X - the number of the most common fittings from taken 4, meaning the value of X - We know the value of X by the formula (1):

In this order, the law of distribution of the value of X is the number of other fittings from taken 4:

To re-verify the correctness, let's revise it, why do the sum of the imaginary

Suggestion: Law rozpodіlu

Example 2: Zastosovuvannya method likuvannya to produce before the dressing in 95% of the cases. Five sick people zastosovuvali tsey method. To know the largest imovirne number was given, as well as the very numerical characteristics of the dropping value X - the number was dressed for 5 ailments, so the given method was put in place.

The theory of imovirnosti is invisibly present in our life. We are not afraid of respect, but the skin of our life can have that chi іnshu ymovіrnіst. Taking up to respect the impersonal options for the development of podia, it becomes necessary for us to designate the least movable and the least movable from them. It is the most convenient way to analyze such imovirnist data graphically. For whom you can help rozpodіl. Binomial - one of the easiest and most accurate.

The first one is to go straight to mathematics and that theory of imovirnosti, let’s consider it, whoever first foresaw such a kind of subdivision of that kind of history of the development of a mathematical apparatus for that understanding.

History

The understanding of imovirnosti in the house is still from the old hours. However, the old mathematicians did not give it any special significance and could only lay the foundation for the theory that has now become the theory of imovirnosti. The stench created deyakі combinatorial methods, yakі greatly helped them, who later created that having developed the theory itself.

In the other half of the seventeenth century, it began to form the main ones to understand that method in the theory of immovability. Bulo introduced the designation of fallow values, ways of calculating the ymovirnosti simple and some folding independent and fallow deposits. Dictations are such an interest to the vapadkovy magnitudes and imovirnosti buv gambles: a skin person wanted to know if she had a chance to win over grі.

The next step was the development of the theory of the imovirnosti of the methods of mathematical analysis. Tsim was engaged in mathematicians such as Laplace, Gauss, Poisson and Bernoulli. The stench themselves slipped the chiu bodice of mathematics onto new rіven. James Bernoulli, himself, debunked the law of binomialism. Before the speech, as we know it later, on the basis of this opinion, another sprat was crushed, as if they were allowed to make the law normal rozpodіlu and many more.

At once, first of all, I began to describe rozpodіl bіnomny, mi trohi osvіzhim in the memory of understanding the theory of imovіrnosti, singly already forgotten zі shkіlnoї lava.

Fundamentals of the theory of imovirnosti

Let's look at such a system, as a result of which there are only two possible results: "success" and "not success". It’s easy to understand on the butt: we give a coin, guessing that it’s tails. Imovirnosti kozhnoї s possible podia (vipade tails - "success", vipade eagle - "not successful") is better than 50 thousand square meters with an ideal balance of money and the presence of other factors, which can be included in the experiment.

Tse bula was the simplest podia. Ale buvayut th folding systems, In which the last steps are counted, and the results of these actions are taken into consideration. For example, let's take a look at such a system: at the box, instead of which we cannot look at it, there are six absolutely identical bags, three pairs of blue, red white tickets. We are to blame for the misfortune of navmannya kіlka sack. Apparently, having pulled one of the big bags with the first one, we will change in the razor-sharpness of the fact that we will step on the bag. It depends on the fact that the number of objects in the system changes.

At the advancing razdіlі razglyadnіshі folded mathematical understanding, which vpritul lead us to what the words "normal rose" mean, "bіnom_alny rose" just.

Elements of mathematical statistics

In statistics, as one of the spheres of the theory of immovability, there are impersonal applications, if the data for analysis is not given in an obvious way. Tobto not in the numerical, but in the sight of the bottom behind the signs, for example, behind the articles. In order to get to such data the mathematical apparatus and to rob the results like vysnovka, it is necessary to convert the output data to a numerical format. As a rule, for a given positive result, a value of 1 is assigned, and a negative result is assigned a value of 0. In this way, we take statistical data, as it is possible to apply analysis for the help of mathematical methods.

The offensive term for the rozumіnnі moreover, scho such a bіnomny rozpodіl vypadkovoї value, - tse vyznachenny variance vypadkovy value and mathematical ochіkuvannya. Let's talk about it at the offensive division.

Mathematical refinement

Really understand that such mathematical perfection is awkward. Let's take a look at the system, de іsnuє a lot of different podіy zі і іїmіmi іmоіrnosti. Mathematical ochіkuvannyam is called the value, which is the sum total of the values ​​of these subdivisions (and the mathematical one, about which we were told in the past, was divided) on the university of their creations.

Mathematically, the calculation of the binomial distribution is subdivided according to the same scheme: we take the value of the variable value, multiplying it by the positive result, and then subsuming the data for all values. It’s even easier to visualize the numbers of data graphically - it’s easier to understand the difference between mathematical estimates of different values.

At the attacking razdіlі mi rozpovіmo you trohi about іnshe understand - the variance of the magnitude. It is also closely related to such concepts, as a bіnomny rozpodіl ymovіrnosti, and є yogo characteristic.

Dispersion of the binomial distribution

Qia value is closely related to the front and also characterizes the distribution of statistical data. Vaughn is the middle square of the given value in terms of their mathematical estimation. That is why the dispersion of the vertical value is the sum of the squares of the differences between the values ​​of the vertical value and the її mathematical points, multiplied by the sum of the sum of the squares.

All that we need to know about dispersion is buried in order to understand that such a bіnomne rozpodіl іmovіrnosti. Now let's move on without intermediary to our topics. And to the point of crying for such a seemingly foldable word-finding "the binomial law has been subdivided."

Binomial rozpodil

Let's figure it out for the cob, why did this rozpodіl binomial. It looks like the word "binom". Possibly, if you think about Newton's binomial - such a formula, for an additional one, you can lay out the sum of two numbers a and b be some kind of invisible world n.

As you, singly, already guessed, Newton's binomial formula and the formula of the binomial subdivision are practically the same formulas. Behind this is more than a blame, because there is no other applied value for specific quantities, and the first one is only a wild mathematical tool, which can be practically used for different purposes.

Formulas rozpodіlu

The function of the binomial subdivision can be written in the view of the sum of the advancing terms:

(n!/(n-k)!k!)*p k *q n-k

Here n is the number of independent vipadical experiments, p is the number of distant observations, q is the number of recent observations, k is the number of the experiment (you can take values ​​from 0 to n),! - the value of the factorial, such a function of the number, the value of which is more expensive for the completion of all numbers that go to it (for example, for the number 4: 4!=1*2*3*4=24).

Incidentally, the function of a binamic subdivision can be written in a seemingly irregular beta function. However, it’s already more foldable, as if it’s less likely to win at the variance of foldable statistical tasks.

The bionomial rose, applied to what the world has seen more, is one of the simplest types of roses in the theory of imaginary. It is also normal rozpodil, which is one of the types of binomial. It is the most victorious, and the simplest among the roses. Bouvai also ripped Bernoulli, ripped Poisson, smarter ripped. All stinks characterize graphically the areas of imovirnosti of that chie process for different minds.

In the offensive division, we can look at the aspects of zastosuvannya of the mathematical apparatus in real life. At first glance, it seems obvious that this is a damn mathematical thing, like, for sure, you don’t get stuck in real life, and it’s not necessary for anyone, let’s look at the mathematicians themselves. However, that is far from the case. Even so, all kinds of rozpodіlіv those їkh graphic manifestations were created exclusively for practical purposes, and not like a baganka of vchenih.

Zastosuvannya

Insanely, the most important stosuvannya rozpodіlu know from the statistics, even if there is a need for a complex analysis of a lot of data. As practice shows, even a lot of arrays of data may have approximately the same values: critical areas are less than low and more than high values, as a rule, there are fewer elements, lower than the average value.

Analysis of the great arrays of data is necessary for statistics. Wine is indispensable, for example, in physical chemistry. In this science of wines, there are signs of richness of values, connected with the fluctuations and displacements of atoms and molecules.

At the attacking division, we understand how important it is to understand such statistical ones, as a bionom rozpodіl vipadkovoї size in everyday life for us and you.

What do I need?

It’s rich to put one’s own food, if mathematics is on the right. And among others, mathematics is not without reason called the queen of sciences. Vaughn is the basis of physics, chemistry, biology, economics, and in the skin sciences zastosovuetsya, among other things, whether it’s a rozpodil: whether it’s a discrete bіnomny rozpodіl, chi is normal, it’s not important. And if we are more often surprised to the navkolyshny world, it’s more likely that mathematics zastosovuetsya skryz: in everyday life, on robots, and inspire people’s eyes, you can pay taxes on looking at statistical data and conduct their analyzes (so, before speech, from special organizations that collect information).

At once we’ll talk a little about those who are robiti, as you need to know more on this topic, lower those that we have mentioned in this article.

That information, as we gave in these articles, is far from true. Іsnuє impersonal nuances in order to, as a form, you can fill it up. The bionomial rozpodil, as we have already mentioned, is one of the main types, on which all mathematical statistics and the theory of imovirnosti are based.

As soon as you feel like crying, you need to know more about this topic, you will need to learn more specialized literature. Start following the university course mathematical analysis and go to the division of the theory of imovirnosti. Also, knowledge in the field of rows is needed, even if it is a bіnomny rozpodіl ymovirnosti - price no more, as low as the last members.

Visnovok

First, finish the article, we would like to tell you one more cicava river. Vaughn stosuetsya without intermediary those of our stats and all mathematics in the wild.

A lot of people are being told that mathematics is a marna science, and they didn’t need anything that stink was in school. But we won’t take knowledge in any way, and even though you didn’t need it in life, it means that you just don’t remember it. If you have knowledge, stench can help you, but if you don’t know, then help them not to be checked.

Otzhe, we looked at the understanding of the bіnomnogo rozpodіlu and everything related to him, and talked about those who are stuck in our life with you.

In classical statisticians, for estimating the value of a binomial distribution, it is often recommended to win formulas that are based on boundary theorems (like the Moivre-Laplace formula). It is necessary to note that from the day of counting points The value of these theorems is close to zero, especially at once, if it is practical to have a hard computer on the skin table. The main shortcoming of inducing approximations is their obvious lack of accuracy at values ​​of n that are characteristic of large additives. Not less than a small amount of time, there is no doubt that there should be clear recommendations on how to fix these and other approximations (in standard texts, more asymptotic formulations are induced, they are not accompanied by accuracy estimates and, therefore, there is little color). I would say that offensive adjective formulas are less for n< 200 и для совсем грубых, ориентировочных расчетов, причем делаемых “вручную” с помощью статистических таблиц. А вот связь между биномиальным распределением и бета-распределением позволяет вычислять биномиальное распределение достаточно экономно.

I don’t see here the task of searching for quantiles: for discrete roses, it’s trivial, but for quiet tasks, they blame it, there, ring out, and it’s relevant. If you still need a quantum, I recommend that you reformulate the task in such a way that it works with p-values ​​(preservation of significance). Axis butt: when implementing such enumeration algorithms on the skin, it is necessary to reconsider the statistical hypothesis about the binomial vipadical value. It is necessary to calculate the statistics according to the classical approach on the skin crotch according to the criterion and the difference between the values ​​of the critical multiplier. Oskіlki, however, the algorithm of enumerations, be brought to the point between the critical multiplier anew (eventually from short to short, the selections are changed), which unproductively increases the time spent. The current pidkhid recommends that you calculate the cautionary significance and balance it with trustworthiness, foreseeing the quantiles.

Therefore, in the codes below, the calculation of the reversal function is indicated, the function rev_binomialDF is indicated, as it calculates the success rate in the case of a given number of tests for a given number of tests, the number m of successes in them is the value of the tests. At the same time, there are vyschezvaniyas vyschezdyazok links between binomial and beta-rozpodil.

In fact, this function allows you to trim between trusted intervals. It’s true, let’s say that in n binomial trials we got m successes. Apparently, the left margin of the two-sided confidence interval for the parameter p with the confidence equal is equal to 0, so m = 0, and є the solutions equal . Similarly, the rights of the border are equal to 1, so m = n, and є decisions are equal . Zvіdsi screaming, scho for a joke of the left cordon, we can virishuvati shkodo rivnyannia , and for a joke of the right - equal . The stench is broken in the functions binom_leftCI and binom_rightCI , which rotate the upper and lower inter-two-way confidence interval in a correct way.

I would like to respect that even though we don’t need the same name accuracy, then with great n you can speed up with an offensive approximation [B.L. van der Waerden, mathematical statistics. M: ІL, 1960, goal. 2, sec. 7]: , de g - quantile of the normal distribution subdivision The value of the approximation is that it is even simpler approach, which allows you to calculate the quantile of the normal distribution subdivision (div. text about the calculation of the normal distribution subdivision and the relative distribution of this indicator). In my practice (mainly, with n > 100), this approximation gave approximately 3-4 characters, which, as a rule, is enough.

To calculate additional lower codes, you will need the files betaDF.h , betaDF.cpp (div. extension about beta-rozpodil), as well as logGamma.h , logGamma.cpp (div. addendum A). You can also marvel at the example of some of the functions.

binomialDF.h file

binomialDF.cpp file

Let's take a look at the binomial rozpodil, let's calculate the mathematical grading, variance, fashion. For additional functions of MS EXCEL BINOM.DIST(), we will need the graphs of the function to expand the scope and flexibility. Let's estimate the parameter rozpodіlu p, mathematical scaling rozpodіlu that standard vodhilennia. So Bernoulli razglyanoli.

Appointment. Let it be held n test, in skin problems, there may be only two subdivisions: subdivision "success" p or podia "failure" with ymovirnistyu q =1-p (so called Bernoulli scheme,Bernoullitrials).

Іmovirnіst otrimannya rіvno x success at tsikh n trying out more:

Number of successes in the selection x є vipadkovoy size, yak maє Binomial rozpodil(English) Binomialdistribution) pі n – є the parameters of which rozpodіlu.

Guess what for zastosuvannya Bernoulli schemes and apparently Binomial rozpodіlu, owe buti vikonanі so think:

• skin testing is due to the mother, there are exactly two results, which are mentally called “success” and “failure”.
• the result of skin testing is due to the fall in the results of previous tests (independence of testing).
• Imovirnist success p can be fast for everyone to try.

Binomial rozpodіl in MS EXCEL

For MS EXCEL, starting from version 2010, for є function BINOM.DIST() , English name - BINOM.DIST() X"Success" (tobto. function p(x), div. higher formula), i the integral function was subdivided(imovirnіst of what the vibіrti will have x or less "success", including 0).

Prior to MS EXCEL 2010 EXCEL, the BINOMDIST() function was available, as it also allows you to calculate rozpodіlu functionі strength and mobility p(x). BINOMDIST() is omitted in MS EXCEL 2010 for consistency.

Attached file has graphics Gustini rozpodіlu ymovіrnostіі .

Binomial rozpodil may be recognized B (n ; p) .

Note: To wake up integral function rozpodіlu ideally fits diagram type Schedule, for Gustini rozpodila – Histogram with groupings. Read the article about the main types of diagrams for a report about the prompt diagram.

Note: For clarity of writing formulas, names for parameters have been created in the file binomial distribution: n and p.

The application file has a different rozrahunki movirnosti for additional functions of MS EXCEL:

As seen in the picture, it is conveyed that:

• In infinite marrow, for which the vibrator struggles, 10% (or 0.1) adnexal elements are removed (parameter p, Third function argument = BINOM.DIST() )
• To calculate the imovirnіst of the one that the vibrator has 10 elements (parameter n, another argument of the function) there will be 5 subordinate elements (the first argument), you need to write down the formula: =BINOM.DIST(5, 10, 0.1, BULLSHIT)
• Remaining, fourth element, insertion = FALSE, tobto. function values ​​are rotated Gustini rozpodila .

If the value of the fourth argument = TRUE, then the BINOM.DIST() function rotates the value integral function rozpodіlu or just The function was subdivided. In this way, it is possible to develop the imovirnist of the fact that the choice of the number of accessory elements will be from the singing range, for example, 2 or less (including 0).

For whom it is necessary to write down the formula: = BINOM.DIST(2, 10, 0.1, TRUE)

Note: With a small value of x, . For example, such a formula to rotate one of the same values: =BINOM.DIST( 2 ; ten; 0.1; TRUE)=BINOM.DIST( 2,9 ; ten; 0.1; TRUE)

Note: Attached file strength and mobilityі rozpodіlu function also calculate the number of assignments for that function NUMBERCOMB() .

Pokazniki rozpodіlu

At file priklad on arkushi buttє formulas for rozrahunka deyah pokanikіv rozpodіlu:

• =n*p;
• (square of standard ventilation) = n * p * (1-p);
• = (n + 1) * p;
• =(1-2*p)*QUEEN(n*p*(1-p)).

Let's get the formula mathematical refinementbinomial distribution, vicorist Bernoulli scheme .

For the purpose of the vipadka value X in Bernoulli scheme(Bernoulli random variable) may rozpodіlu function :

Whose rozpodil is called rozpodіl Bernoulli .

Note : rozpodіl Bernoulli- okremy vipadok binomial distributionіz parameter n=1.

Generate 3 arrays of 100 numbers with different success rates: 0.1; 0.5 and 0.9. For whom at the vikni Generation of upright numbers let's put in the following parameters of skin condition p:

Note: How to set option Vipadkove rozsiyuvannya (Random seed), then it is possible to choose a different set of generated numbers. For example, by setting this option =25, you can generate the same sets of random numbers on different computers (however, of course, other parameters are different). Option value can accept any number of values ​​from 1 to 32767. Option name Vipadkove rozsiyuvannya you can get lost. Better bulo b її shift the yak Set number with correct numbers .

The result has 3 columns of 100 numbers, on the basis of which you can, for example, evaluate the success p behind the formula: Number of successes/100(Div. file butt sheet GenerationBernoulli).

Note: For rozpodіlu Bernoulli z p \u003d 0.5, you can win the formula \u003d VIPAD МІЖ (0; 1), so you can confirm.

Generation of vipadkovy numbers. Binomial rozpodil

Let's assume that the sample had 7 defective samples. Tse means that the situation is “even better”, that a part of the defective samples has changed p as a characteristic of our brewing process. Although such a situation is “due ymovirna”, but it’s still ymovirnist (alpha-rizik, pardon of the 1st kind, “hibna anxiety”), but still p was left without changes, and the increase in the number of defective samples was determined by the vibrancy of the sample.

As can be seen on the little one below, 7 - the number of defective samples, as it is acceptable for the process p = 0.21 with the same value Alpha. This is an illustration of how the threshold value of defective viruses has been exceeded in the selection, p"Swiss for everything" has increased. The phrase "greater for everything" means that there is only 10% correctness (100%-90%) of the fact that the elimination of a part of defective virobiv more than a threshold viklikana is no more than random reasons.

In this way, the excess of the threshold number of defective viruses in the selection can serve as a signal that the process is embarrassed and becoming about more number of defective samples.

Note: Prior to MS EXCEL 2010, EXCEL had a CRITBINOM() function, which is equivalent to BINOM.OBR(). KRITBINOM is omitted in MS EXCEL 2010 and more for summation.

Link of the Binomial subdivision with other subdivisions

How to parameter nbinomial distribution pragne inconsistency, and p pragne to 0, then in your vipadka Binomial rozpodil can be approximated. You can formulate mind, if you are close rose under Poisson practice good:

• p(the less p and more n, Tim closer);
• p >0,9 (vrakhovuyuchi sho q =1- p, in which case it is necessary to carry out the calculation through q(a X it is necessary to replace n - x). Father, what is less q and more n, Tim closer).

At 0.110 Binomial rozpodil can approximate.

In your hell Binomial rozpodil you can be good neighbors, if the world of marriage N hypergeometric rozpodіlu there is a lot more expansion of the choice of n (tobto, N>> n or n/N More details about the linking of the visualization of roses can be read in the article. Approximations are also made there, and explanations of the mind, if possible, and with some accuracy.

PORADA: You can read about other rozpodіli MS EXCEL at the article.

- (binomial distribution) Rozpodil, which allows the development of the present, as a vipadkovoy under, taken away as a result of a guard to a number of independent under, as a result, the imovirn_st will come, so to become yoga elementary… Economic dictionary

- (determined by Bernoulli) derivation of the number of occurrences of sepsis under the repeated independent testing, as the appearance of positive signs in the skin test is good p(0 p 1). The very number? appearing tsієї podії є… … Great Encyclopedic Dictionary

bіnomny rozpodіl- - Topics of electrical communications, basic concepts EN binomial distribution ...

- (determined by Bernoulli), derivation of the number of appearances of a singal undergrowth during repeated independent tests, as the appearance of a positive test in the skin test is a good one (0≤p≤1). Same, kіlkіst μ appear tsієї podії… … Encyclopedic dictionary

bіnomny rozpodіl- 1.49. Binomial rozpodіl Rozpodіl ymovіrnosti discrete vypadkovoї value X, which accepts whether or not the value of 0 to n, so that when x = 0, 1, 2, ..., n і parameters n = 1, 2, ... і 0< p < 1, где Источник … Glossary of terms for normative and technical documentation

Rozpodil Bernoulli, rozpodіl ymovirnosti vypadkovoї value X, scho takes tsili values ​​with ymovіrnosti vіdpovidno (bіnomіalny kofіtsіent; p parameter B.r., called. Mathematical Encyclopedia

Razpodіl ymovіrnosti kіlkosti appearing sing podії during repeated independent testing. As with skin testing, it appears to be similar, moreover, 0 ≤ p ≤ 1, then μ appears to be positive with n independent ... ... Great Radianska Encyclopedia

- (determined by Bernoulli), deriving the number of occurrences of a certain number of appearances in repeated independent tests, as a result of the appearance of a number of appearances in the dermal examination of a healthy p (0<или = p < или = 1). Именно, число м появлений … Natural science. Encyclopedic dictionary

Binomial rozpodіl ymovіrnosti- (binomial distribution) Rozpodil, which is to be feared in depressions, if the result of a skin independent experiment (statistical caution) takes one of two possible values: overpowering chi damage, including chi vinyatok, plus chi ... Economic and Mathematical Dictionary

bіnomne rozpodіl ymovіrnosti- Rozpodil, which is to be guarded against in depressions, if the result of a skin independent experiment (statistical caution) has one of two possible meanings: the power of the strike, the inclusion of the vine, plus the minus, 0 chi 1. Tobto ... ... Dovіdnik technical translation

Books

• The theory of imovirnosti and mathematical statistics in zavdannyah. More than 360 head and right, D. A. Borzikh. At the proponated helper, there is a task for a different balance of foldedness. Prote the main emphasis is broken down into the middle order. The whole thing was specially cut in order to spur the students to…
• The theory of uniformity and mathematical statistics in the heads of more than 360 heads and the right, Borzikh D.. Prote the main emphasis is broken down into the middle order. The whole thing was specially cut in order to spur the students to…