The theory of virality. Basic terms and understanding. Basic concepts of the theory of properties and mathematical statistics
for 2nd year students of all specialties
Department of Vishchoi Mathematics
Partially available
Shanovni students!
With your respect, we present the overview (installation) lecture of Professor N.Sh. Kremer from the discipline “Theory of Inevitabilities and Mathematical Statistics” for students of another year of VZFED.
The lecture is discussed zavdannya development of the theory of economics and mathematical statistics at an economic university This is the place in the system of training daily economists, what is considered organization independent Works of students with the help of a computer-based initial system (COPR) and traditional aids are given overview of the main provisions this course, as well as methodological recommendations for further training.
Among the mathematical disciplines that are studied in economic universities, the theory of inequalities and mathematical statistics occupy a special position. First of all, it is the theoretical basis of statistical disciplines. In another way, the methods of the theory of natural properties and mathematical statistics are directly used in the treatment of mass aggregates monitoring of symptoms, sampling the results of monitoring and identifying patterns of epileptic symptoms. It turns out that the theory of probability and mathematical statistics are of more important methodological significance cognitive process, with revealed hidden patterns follow-up processes, serve as a logical basis inductive-deductive logic.
Kozhen, a student of another year, is guilty of an offensive set (case) from the discipline “Theory of Intellectualities and Mathematical Statistics”:
1. Oglyadov's mentored lecture with this discipline.
2. Podruchnik N.Sh. Kremer’s “Theory of Virtues and Mathematical Statistics” – M.: UNITI – DANA, 2007 (later simply called “handyman”).
3. Basic methodological guide“The theory of natural values and mathematical statistics” / ed. N.Sh. Kremer. – M.: Vuzovskiy podruchnik, 2005 (also referred to as “posіbnik”).
4. Computer basic program COPR with discipline (later – “computer program”).
On the Institute’s website, on the “Corporate Resources” page, there is an online version of the computer program KOPR2, a survey lecture and an electronic version of the handbook. In addition, the computer program and reference book are presented on CD - ROM ah for students of another course. That “paper-looking” student needs a mother’s assistant.
Let us explain the significance of the skin from the initial methodological materials that are included in the assigned set (case).
At the friend's The main provisions of the initial material of the discipline are outlined, which illustrate the achievement of a great number of tasks.
U Pos_bnik methodological recommendations were given for self-learning of the basic material, the most important concepts of the course and typical tasks were seen, control food was given for self-testing in this discipline, options for home control work were given, And the student is to blame for the viconics, as well as the methodical additions to their viconics.
Computer program Called to give you maximum assistance in the acquired course in the regime dialogue programs with the student in order to maximize your attendance in your classroom, including contact with the depositor.
For a student who is starting to use the system of distance learning, the next most important thing is organization of independent work
When starting to learn this discipline, read the entire overview (instructional) lecture to the end. Therefore, I would like to give you a general overview of the basic concepts and methods taught in the course “Theory of Inferences and Mathematical Statistics”, which can be used to prepare VZFED students.
Before skin grafting Become familiar with the methodological recommendations for learning given by those from the guide. Here you will find a selection of basic foods that you can read; Explain what concepts, meanings, theorems, knowledge are the most important, what we need to learn and master beforehand.
Then move on to education basic elementary material following the guide is consistent with the removal of methodological recommendations. It’s good to outline the main meanings, formulation of theorems, schemes of their proofs, formulas and most typical tasks. The formulas must be written in a special table of the skin part of the course: theory of validity and mathematical statistics. Regularly taking notes, looking at tables of formulas, improves their memorization.
After processing the main initial material with the skin using a tool, you can proceed to processing it using an additional computer initial program (KOPR2).
Increase respect for the structure of everyday computer programs with skin issues. After naming those, a transfer of the main initial meals will be made to those following the instructions from the designated numbers of paragraphs and pages that need to be read. (It is likely that the transfer of this nutrition from the skin was also caused by the companion).
Then, in the short form, preliminary material is given with these topics (and with adjacent paragraphs with these topics) - the main meanings, theorems, power and signs, formulas, etc. During the process of studying topics, you can also click on the screen those fragments of preliminary material (besides or previous topics) that are needed at the moment.
Then you are introduced to the basic material and basic standard assignments ( apply it), Most of them can be seen in the mode dialogue program for a student. The functions of a number of applications are interspersed with displays on the screen followed by the stages of the correct decision to begin. During the process of looking at most applications, you will be given nutrition of a different nature. How to enter data on one power supply from the keyboard digital proof, on other – choose the correct line(s) with a number of registered ones.
Once you have entered the information, the program confirms its correctness or confirms, after reading the hint, to replace the necessary theoretical positions, try again the dates of the correct solution that's the truth. In many tasks, the number of solution trials has been set to limit the number of solution attempts (if the solution is moved, the correct solution will be displayed on the screen). And such applications, which have a lot of information that can be contained in the prompt, are increasing in the world of repetition of recent tests of the same type.
After familiarizing yourself with the theoretical principles of the initial material and the practicalities that the report analysis of the decision entails, you have the right to exercise self-control in order to consolidate the skills of most typical skin tasks. The task of self-control also reduces elements of dialogue with the student. Once you have completed your solution, you can check the correct answer and match it with the one you gave.
Upon completion of work with skin traces, the control department was completed. The correct answers will not be displayed to you, and your answers will be recorded on your computer’s hard drive for the consultant (tutor) to read about.
After completing topics 1–7, you are responsible for defeating home control robot No. 3, and after completing topics 8–11 – home control robot No. 4. Options for assigning control robots are indicated to your assistant (his electronic version). The number of the option that is added must coincide with the remaining digit of the number of your special certificate (lease book, student certificate). For each control robot, you are required to undergo a conversation in which it is necessary to show the correct knowledge and knowledge of the basic understanding (values, theorems (without proof), formulas, etc.) on the topic of the control robot. The discipline is completed with a course examination.
The theory of natural phenomena is a mathematical science that studies the patterns of epileptic phenomena.
Proposed for advanced training, the discipline consists of two sections: “Theory of Inevitability” and “Mathematical Statistics”.
Mathematics includes several areas, including algebra and geometry, and the theory of certainty. Find out the terms that are hidden in all these directions, and besides them, they are specific and have power over just one specific “niche” word, formula, theorem.
The coined word “theory of authenticity” makes an unprepared student panic. True, there are clearly pictures that depict terrible volumetric formulas, and the most important task alone takes up a whole lot. However, in practice, everything is not so greedy: it’s enough to understand the meaning of certain terms once and get into the essence of some kind of logic of mercilessness, so that you stop being afraid of the same thing again and again. In connection with this, we will look at the basic concepts of the theory of probability and mathematical statistics - young, or at the edge, knowledge.
Please understand
The function of the language is to convey information from one person to another so that they understand, are informed and can be corrected. This mathematical concept can be explained in simple words, but in this case the act of exchanging data took significantly more than an hour. You will realize that instead of the word “hypotenuse” you would soon have to say “the found side of the recticutaneous tricutaneous” - but it is extremely clumsy and time-consuming.
That is why people come up with new terms for these and other phenomena and processes. The main concepts of the theory of validity - the idea, the validity of the concept, etc. - appeared just like that. This means that in order to master formulas, figure out knowledge and consolidate skills in life, it is necessary not only to memorize new words, but also to understand what it means to use them. The more you become aware of them, the more you delve into the situation, the wider the scope of your possibilities becomes, and the more you absorb more light.
Why does the subject have a sense?
Known from the basic concepts of the theory of validity. The classic meaning of reliability sounds like this: the goal is to control the results of the investigation to a minimum number of possible ones. Let’s look at a simple example: if a person rolls a dice, her skin may fall out of six sides of the fire. Thus, the total number of results is six. The probability of the fact that the side is double-sided is 1/6.
It is important to convey the same result, which is extremely important for various facists. How many defective parts will be found in the batch? How much to lay aside, how much to earn. What kind of confidence can help heal the ailment? Such information is vitally important in everyday life. Let’s not spend an hour on additional applications and start learning a new area.
First knowledge
Let's take a look at the basic concepts of the theory of authenticity and their derivatives. In law, natural sciences, economics, lower formulas and terms are used everywhere, since they may be directly related to statistics and extinction. The report on the development of this nutrition will reveal to you new formulas for more accurate and simple calculations, which will be done in a very simple way.
One of the most basic and fundamental to understand the theories of probability and mathematical statistics is the fallacy theory. Explained in sensible words: from all the possible results of the experiment, only one is avoided as a result. It is clear that the current situation means that the other thing will be different, although theoretically the result may be different.
Since we conducted a series of experiments and collected a number of results, then the skin's validity is calculated using the formula: P(A) = m/n. Here m is the number of times we tried the series to prevent the appearance of the desired result for us. There is a limited number of experiments that have been carried out. If we tossed a coin 10 times and got heads 5 times, then m=5 and n=10.
Look at it
It is clear that the same result is guaranteed to be true when tested on the skin - this method is called reliable. If there is no way to exist, then it is called impossible. However, such ideas do not stagnate in the minds of the theory of reliability. The basic concepts that are much more important to know are the most important and absurd ideas.
It is ensured that by the time the experiment is carried out, two stages are simultaneously formed. For example, we are rolling two dice - the fact that one roll shows “six” does not guarantee that the other roll doesn’t come up with a different number. Such pods will be called sleeping.
If we roll one die, two numbers cannot roll up at the same time. In this case, the results of “one”, “two”, etc. will be considered as nonsense. It is important to analyze what results may be present in each skin condition - where to find what formulas to use in the task of finding balances. We will continue to summarize the basic concepts of the theory of intelligibility through a number of paragraphs, if we look at the peculiarities of the folding and multiplication. Without them, life will never be possible.
Suma and tvir
Let’s say you roll a dice with a friend, and he gets a few dice. To overcome this, you need to cut off five or six. The odds of getting both numbers will be 1/6, so it looks like 1/6 + 1/6 = 1/3.
And now you know that you roll the two dice, and your friend takes away 11 points. Now you need the two of you to get “six” right away. The two are independent, so you need to multiply: 1/6*1/6 = 1/36.
Among the main things to understand and the theorems of the theory of incompetence, we need to gain respect for the amount of incommensurability of all the worlds, such as those that can arise overnight. The formula added in this way is as follows: P(A+B) = P(A) + P(B) - P(AB).
Combinatorics
Quite often we need to know all the possible modifications of certain parameters of an object or calculate the number of combinations (for example, when selecting a cipher). With this we can be helped by combinatorics, which is closely related to the theory of probability. The main concepts here include several new words, and low formulas from which you will need to sing.
Let's say you have three numbers: 1, 2, 3. You need to write all possible three-digit numbers. How many will there be? Subject: n! (the hail sign means factorial). Combinations with a number of different elements (numbers, letters, etc.), which vary only in the order of their arrangement, are called permutations.
However, we often encounter this situation: there are 10 digits (ranging from zero to nine), which form a password or code. Let’s say that this dovzhina is 4 characters. How can I unlock the number of possible codes? And this is why there is a special formula: (n!)/(n - m)!
The doctor is assigned a higher mental task, n=10, m=4. Then you will need more than simple mathematical developments. Before speaking, such combinations will be called placed.
By the way, the basic concept is understood - there are sequences that are divided into one type of one or another element. Their number is calculated using the formula: (n!)/(m!(n-m)!).
Mathematical calculation
It is important to understand what a student faces during his first studies in the subject, i.e. mathematical education. This is the sum of all possible resulting values, multiplied by its validity. In essence, this is the average number that we can transfer as a result of testing. For example, there are three values for which weights are indicated: 0 (0.2); 1 (0.5); 2 (0.3). In a mathematical sense: M (X) = 0 * 0.2 + 1 * 0.5 + 2 * 0.3 = 1.1. In this way, from the protonated virus it is possible to ensure that the given value is constant and does not lie in the result of testing.
This concept is illustrated in many formulas, and you will come across it more than once in the future. It is difficult to work with him: the mathematical calculation of the sum is the same as the ancient sum. ochikuvan - M(X+Y) = M(X) + M(Y). The same goes for the creation: M(XY) = M(X) * M(Y).
Dispersion
You probably remember from your school physics course that dispersion is dispersion. What is the best place to understand the theory of virtuosities?
Marvel at the two butts. In one round we are given: 10 (0.2); 20(0.6); 30(0.2). Otherwise – 0(0.2); 20(0.6); 40(0.2). It will be possible to understand mathematically in both cases, however, how can we compare these situations? It’s also true that the difference in value for the other type is significantly greater.
And therefore the concept of dispersion was abandoned. To eliminate this, it is necessary to develop a mathematical calculation based on the sum of the difference between the skin drop value and the mathematical calculation. Let's take the numbers from the first butt, written in the first paragraph.
The initial mathematical calculation is: M(X) = 10*0.2 + 20*0.6 + 30*0.2 = 20. The resulting dispersion value: D(X) = 40.
Another important way to understand statistics and the theory of validity is mean square variation. It’s really easy to figure it out: you just need to take the square root of the variance.
Here we can use such a simple term as scope. This value means the difference between the maximum and minimum values of the sample.
Statistics
Some basic school concepts are often violated by science. Two of them are the arithmetic mean and median. Chantly, you remember how to know their meanings. Let’s remember about every drop: the arithmetic mean is the sum of all values, divided by its totality. If the value is 10, we add them up and divide them by 10.
The median is the central value of the middle of all possible ones. Since there are an unpaired number of values, we list them in increasing order and select those that appear in the middle. Since we have a number of values, we take the central two and divide by two.
Two more values that vary between the median and two extreme values - the maximum and minimum - are called quartiles. They are calculated this way - if the number of elements is unpaired, the number that appears in the middle of the row is taken, and if the number is paired, half the sum of the two central elements is taken.
There is a special graph on which you can calculate all sample values, including range, median, quarterly interval, as well as values that are not included in the statistical data. The image that comes out has a very specific (and actually non-mathematical) name - “a box of hair.”
Rozpodil
Rozpodil also gained a basic understanding of the theory of probability and mathematical statistics. In short, here is detailed information about all the variable values that we can obtain as a result of testing. The main parameter here will be the possibility of the appearance of a specific skin value.
A normal division is one in which there is one central peak, which contains the value that sharpens most often. Fewer and lesser results diverge from each other. The graph from the side looks like a “girka”. Next, you will recognize that this type of division is closely related to the central boundary theorem, which is fundamental to the theory of reliability. She describes regularities that are important for the branch of mathematics we are considering, even in different situations.
Let's turn to those. There are two more types of divisions: asymmetric and multimodal. The first one looks like half of the “normal” graph, so the arc descends just one step below the peak value. Let's face it, the multimodal division is the same as the number of “upper” values. The graph goes up and down like this. The highest frequency value in any division is called fashion. This is also one of the most important things to understand about the theories of probability and mathematical statistics.
Gaussian split
Gaussian, or more normally, a division - one in which the vigilance of the average is subject to the singing law.
Briefly speaking, the main difference in the value of the selection is exponentially different from fashion - the most common of them. To be more precise, 99.6% of all values are distributed between three standard values (remember, we looked at this concept more?).
The Gaussian division is one of the main things to understand in the theory of stability. Beyond this, you can understand that an element beyond these and other parameters is included in the category of “typical” - this is how the growth and vitality of a person is assessed according to age, the level of intellectual development, the psychological state and much more.
Yak zastosuvati
It is worth noting that “boring” mathematical data can be taken into account for its own sake. For example, one young man used the theory of odds and statistics to win millions of dollars at roulette. True, before this I had to prepare – for many months I had to record the gambling results at different casinos.
After carrying out the analysis, we found that one of the tables had a small amount of damage, which means that a number of values are statistically significantly more frequent than the others. There are quite a few rozrakhunki, patience - and the rulers of the world are racking their brains, thinking how people can be so kind.
The impersonality of everyday everyday tasks, which is impossible to figure out without looking at statistics. For example, how do you know how many stores can supply clothes in different sizes: S, M, L, XL? For this purpose, it is necessary to analyze who often buys clothes locally, in the area, in nearby stores. If such information is not rejected, the ruler risks spending a lot of money.
Visnovok
We looked at some of the most important theories of universality: sampling, variation, permutation and placement, mathematical calculation and dispersion, mode and normal distribution... In addition, we looked at a number of formulas, which are based on it usually takes more than a month.
Don't forget: mathematics is necessary for economics, natural sciences, information technologies, and engineering specialties. The statistics of one of these regions also cannot be summarized here.
Now on the right side for malim: practice, master the mastery and apply it. The basic concepts and the important theory of authenticity will be forgotten unless time is taken to repeat them. In addition, future formulas will significantly expand on those that we have considered. Therefore, try to remember them, especially since they are not so many.
Fundamentals of the theory of virtualities and mathematical statistics
Fundamentals of the theory of probability and mathematical statistics Basic concepts of the theory of probability The subject of the theory of probability is the common pattern of similar phenomena of a mass nature. Meaning 1. Any possible fact about which one can say that will or will not happen in these minds is called. butt. The ampoules that come off the conveyor can be either standard or non-standard. One (even) result from the meaning of two possible ones is called a conclusion. There are three types of hypotheses: reliable, impossible and anecdotal. Significance 2. Reliable is called that idea, because it is impossible to stop working on the development of active minds, then. It's going to get tough. butt. If the urn contains only white bags, then the bag taken from the urn will be completely white. In these minds, the fact of the appearance of a white cheek will be a reliable assumption. Significance 3. It is impossible to call that thing that cannot stand for the development of active minds. butt. It is not possible to pull the white bag from the urn in order to remove the black bag. In our minds, the fact of the appearance of white cheeks will be difficult to understand. Significance 4. Vipadkovym is called the idea that in these very minds may or may not happen. butt. A coin thrown into the air can end up in such a way that either a coat of arms or a number appears on its top side. Here the beast appears, and the other side of the coin also appears in the same way. Meaning 5. Testing - the totality of these minds or actions that can be repeated countless times. butt. Throwing a coin on top is a test, but the result is likely. There is a coat of arms on the top side of the coin, and the numbers are underneath. Significance 6. The possibilities are such that with any given test, there may be only one of them and every other that is not included in the totality, and these ideas are called equally possible. butt. In the urn lie white and black stones and everyday others. Taken on cue, one bag may appear white or black. These ideas are also possible, because The appearance of coolants of other barbaration when tried is turned off. Significance 7. Two conditions A and B are called unreasonable, because in which the smells tested cannot exist at the same time. butt. The coat of arms and the number are the only possible and absurd options for a one-time toss of a coin. Meaning 8. Two species A and B are called mature when tested, since the appearance of one of them does not exclude the possibility of the appearance of another species during the same trial. butt. It is possible for the eagle to appear and the numbers to appear with one toss of two coins. Meaning 9. The steps A i are called equal in this test, because due to the symmetry of the base it is important that each of these steps is not more comparable to the others. butt. Any edge will appear with one throw of the brush and the same shape (behind the wash, as the brush is cut from the same material and takes the shape of a regular hexagon). Significance 10. The pods are called friendly (friendly) singing pods, because the appearance of one of these pods is followed by the appearance of a given pod. The occurrences that turn off the appearance of the species are called unfavorable for this species. butt. The urn has 5 white and 7 black bags. When you take one hand by surprise, either a white or black hand may appear in your hands. In this period, the appearance of a white shoot will result in 5 drops, and the appearance of a black shoot will result in 7 drops, out of a total of 12 possible drops. Meaning 11. Two equally possible and absurd ideas can be called protégés one. If one of these steps is designated A, then the opposite section is designated by the symbol Ā. butt. Hit or miss; A win and a program after a lottery slip – all for the same reason. Meaning 12. If, as a result of any mass operation, which consists of n similar single traces or caution (testing), the pattern of the episodic procedure will appear m times, the number m is called the frequency of the episodic podіi, and the m/n relation is called yogo frequency. butt. Among the first 20 germs that came off the conveyor, 3 non-standard germs (slubs) appeared. Here the number of samples n = 20 the frequency of the sample m = 3 the frequency of the sample m / n = 3/20 = 0.15. Whatever the case, in given minds may have its own objective possibility of appearing, and in some cases this possibility appears more, in others less. For a quick leveling between each other, according to the level of the possibility of their attack with a skin rash, tie a valid number that expresses the level of the objective stage Possibility of the present day. This number is called the balance of power. Meaning 13. The validity of this idea is the numerical measure of the objective possibility of presenting this idea. Value 14. (Classical value). The probability of the event is the ratio of the number m of occurrences favorable to the onset of this event to n of all possible occurrences, then. P(A) = m/n. butt. The urn contains 5 white and 7 black bags, thoroughly mixed. What is the certainty of the one who is taken as a lure from the ballot box and one bag will appear white? Decision. In total, 12 possible episodes have been tested, of which 5 are associated with the appearance of white spots. Therefore, the homogeneity of white spots appears P = 5/12. Value 15. (Statistical value). In order to achieve a large number of repeated tests of a hundred-song song A, if it is noted that the song fluctuates around a certain constant number, then the song A has a consistency of P(A), approximately equal to the frequency, then. P(A)~ m/n. Part of the analysis with an unlimited number of trials is called statistical accuracy. The main power of the world. 1 0 If the effect A causes the effect B (A B), then the validity of the effect A does not outweigh the validity of the effect B. P(A)≤P(B) 2 0 Both the effects A and B are equally strong (A B, B A , B=A), their balance is equal to P(A)=P(B). 3 0 Immovability of any kind can be a negative number, then. P(A)≥0 4 0 The reliability of the reliable method is higher than 1. P(A)=1. 5 0 Possibility of the infeasible subdivision is more advanced 0. Р( )=0. 6 0 The consistency of any type of subdivision A is placed between zero and one 0<Р(А)<1 Основные формулы комбинаторики Определение 1 . Различные группы по m предметов, составленные из n однородных предметов ( m , n ), называются соединениями. Предметы, из которых составляют различные соединения, называют элементами. Существует 3 вида соединений: размещения, перестановки, сочетания. Определение 2. Размещениями по m элементов из данных n элементов ( m ≤ n ) называют такие соединения, которые отличаются друг от друга либо самими элементами, либо их порядком. Например, размещениями из трех предметов a , b и c по два будут следующие соединения: ab , ac , bc , ca , cb , ba . Число размещений из данных n элементов по m обозначают символом А n m = n ( n -1)( n -2)·....·( n - m +1). Пример. А 10 4 =10·9·8·7=5040. Определение 3. Перестановками из n элементов называют такие соединения, которые отличаются друг от друга только порядком элементов. Р n =А n n = n ( n -1)( n -2)...·3·2·1= n ! По определению 0!=1. Пример. Р 5 =5!=1·2·3·4·5=120. Определение 4. Сочетаниями из n элементов по m называются также соединения, которые отличаются друг от друга, по меньшей мере, одним элементом и каждое из которых содержит m различных элементов: C n m === Пример. Найти число сочетаний из 10 элементов по четыре. Решение. C 10 4 ==210. Пример. Найти число сочетаний из 20 элементов по 17. Решение. ==1040. Теоремы теории вероятностей Теорема сложения вероятностей Теорема 1 . Вероятность наступления одного какого-либо события из двух несовместимых событий А и В равно сумме вероятностей этих событий Р(А+В)=Р(А)+Р(В ). Пример. В урне 5 красных, 7 синих и 8 белых шаров, перемешанных между собой. Какова вероятность того, что взятый наугад один шар окажется не красным? Решение. Не красный шар - это или белый или синий шары. Вероятность появления белого шара (событие А) равна Р(А)= 8/20 = 2/5. Вероятность появления синего шара (событие В) равна Р(В)= 7/20. Событие, состоящее в появлении не красного шара, означает появление или А или В, т.к. события А и В несовместимы, то применима теорема 1. Искомая вероятность будет равна Р(А+В)=Р(А)+Р(В)=2/5+ +7/20=3/4. Теорема 2. Вероятность наступления одного из двух событий A или B равно сумме вероятностей этих событий минус вероятность их совместного появления P ( A + B )= P ( A )+ P ( B )+ P ( AB ). Теорема умножения вероятностей Определение 1. Два события A и B называются независимыми друг от друга, если вероятность одного из них не зависит от наступления или ненаступления другого. Пример. Пусть A - событие, состоящее в появлении герба при первом бросании монеты, а B - событие, состоящее в появлении герба при втором бросании монеты, то события A и B не зависят друг от друга, т.е. результат первого бросания монеты не может изменить вероятность появления герба при втором бросании монеты. Определение 2. Два события A и B называются зависящими друг от друга, если вероятность одного из них зависит от наступления или ненаступления другого. Пример. В урне 8 белых и 7 красных шаров, перемешанных между собой. Событие A - появление белого шара, а событие B - появление красного шара. Будем брать из урны наугад два раза по одному шару, не возвращая их обратно. До начала испытания вероятность появления события A равна P ( A )=8/15, и вероятность события B равна P ( B )=7/15. Если предположить, что в первый раз был взят белый шар (событие A ), то вероятность появления события B при втором испытании будет P ( B )=7/14=1/2. Если в первый раз был взят красный шар, то вероятность появления красного шара при втором извлечении равна P ( B )=6/14=3/7. Определение 3. Вероятность события B , вычисленная в предположении, что перед этим наступило связанное с ним событие A , называется условной вероятностью события B и обозначается PA ( B ). Теорема 3 . Вероятность совместного наступления двух зависимых событий ( A и B ) равна произведению вероятности одного из них на условную вероятность другого, вычисленную в предположении, что первое событие произошло, т.е. P ( AB )= P ( A )· P A ( B )= P ( B )· P B ( A ). Теорема 4. Вероятность совместного наступления нескольких зависимых событий равно произведению вероятности одного из них на условные вероятности всех остальных событий, вычисленные в предположении, что все предыдущие события уже наступили: P(A 1 A 2 A 3 ...A k )=P(A 1 )·P A1 (A 2 )·P A1A2 ·P(A 3 )...·P A1A2…A k-1 (A k ) Теорема 5 . Вероятность совместного наступления двух независимых событий A и B равна произведению вероятностей этих событий P ( AB )= P ( A )· P ( B ). Теорема 6 . Вероятность совместного наступления нескольких независимых событий A 1 , A 2 , ... A k равна произведению их вероятностей, т.е. P ( A 1 A 2 ... A k )= P ( A 1 )· P ( A 2 )·...· P ( A k ). Пример. Два стрелка делают одновременно по одному выстрелу в одну цель. Какова вероятность того, что оба попадут, если известно, что первый стрелок в среднем дает 7 попаданий, а второй 8 попаданий на каждые 10 выстрелов? Какова вероятность поражения мишени? Решение. Вероятность попадания первого стрелка (событие A ) равна P ( A )=0,8, вероятность попадания второго стрелка (событие B ) равна P ( B )=0,7. События A и B независимы друг от друга, поэтому вероятность совместного наступления этих событий (совместное попадание в цель) найдем по теореме умножения для независимых событий: P ( AB )= P ( A ) P ( B )=0,8·0,7=0,56. Вероятность поражения мишени означает попадание в мишень хотя бы одного стрелка. Так как попадание в мишень первого и второго стрелков являются событиями совместными, то применение теоремы сложения вероятностей для совместных событий дает следующий результат: P(A+B)=P(A)+P(B)-P(AB)=P(A)+P(B)-P(A)·P(B)=0,8+0,7- 0,8·0,7=0,94. 5.3.3. Формула полной вероятности Определение 4. Если при некотором испытании может произойти одно какое-либо событие из нескольких несовместных A 1 , A 2 ,..., A k , и при этом никаких других событий быть не может, но одно из указанных событий обязательно произойдет, то группу событий A 1 , A 2 ,..., A k называют полной группой событий. Теорема 7. Сумма вероятностей событий, образующих полную группу, равна единице: P ( A 1 )+ P ( A 2 )+...+ P ( A k )=1. Следствие. Сумма вероятностей двух противоположных событий равна единице: P ( A )+ P ( A )=1. Если вероятность одного события обозначим через p , вероятность противоположного ему события обозначим через q , тогда p + q =1. Пример. Вероятность попадания в цель равна 0,94. Найти вероятность непопадания. Решение . Попадание в цель и непопадание являются противоположными событиями, поэтому, если p =0,94, то q =1- p =1-0,94=0,06. Теорема 8 . Если случайные события A 1 , A 2 ... A n образуют полную систему, и если событие B может осуществляться только совместно с каким-нибудь одним из этих событий, то вероятность наступления события B можно определить по формуле: P(B)=P(A 1 )P A1 (B)+P(A 2 )P A2 (B)+...+P(A n )P A n (B) Это равенство называется формулой полной вероятности . Пример. На склад готовой продукции поступили изделия из трех цехов, в том числе: 30% из I -го цеха, 45% из II цеха и 25% из III цеха. Среди изделий I цеха брак составляет 0,6%, по II цеху 0,4% и по III цеху-0,16%. Какова вероятность того, что взятое наугад для контроля одно изделие окажется с браком? Решение. Одно изделие может быть взято или из продукции I цеха (событие A 1 ), или из продукции II цеха (событие A 2 ), или из продукции III цеха (событие A 3 ). Вероятности этих событий будут: P ( A 1 )=0,30; P ( A 2 )=0,45; P ( A 3 )=0,25. Вероятность того, что изделие с браком (событие B ) будет взято из продукции I цеха, есть условная вероятность P A 1 ( B ). Она равна P A 1 ( B )=0,006. Вероятность того, что изделие с браком будет взято из продукции II цеха P A 2 ( B )=0,004 и из продукции III цеха P A 3 ( B )=0,0016. Теперь по формуле полной вероятности найдем вероятность того, что взятое наугад одно изделие будет с браком: P(B)=P(A 1 )P A1 (B)+P(A 2 )P A2 (B)+...+P(A 3 )P A3 (B) = 0,3·0,006+0,45·0,004+0,25·0,0016=0,004. Формула Бернулли Теорема 9. Пусть производится n независимых повторных испытаний по отношению к некоторому событию A . Пусть вероятность появления этого события в каждом отдельном испытании остается неизменно равной p , а вероятность появления противоположного события Ā, есть q . Тогда вероятность появления интересующего нас события A равно m раз при указанных n испытаниях рассчитывается по формуле Бернулли: P m , n = p m q n - m , так как, то P m , n = · p m · q n - m Пример. Коэффициент использования станка в среднем равен 0,8. В цехе имеется 5 станков. Какова вероятность того, что в некоторый момент времени окажутся работоспособными только 3 станка? Решение. Задача подходит под схему повторных испытаний и решается по формуле Бернулли: n =5, m =3, p =0,8 и q =1-0,8=0,2: P 3,5 = (0,8) 3 ·(0,2) 2 =0,2084. Асимптотическая формула Пуассона В статистической практике нередко встречаются такие примеры независимых испытаний, когда при большом числе n независимых испытаний вероятность Р появления события в каждом отдельном испытании оказывается сравнительно малой величиной, стремящейся к нулю с увеличением числа испытаний . При этих условиях для вычисления вероятности Р m , n появление события m раз в n испытаниях пользуются асимптотической формулой Пуассона : Р m,n ≈e -a , где a=np Пример. Доля брака всей продукции завода составляет 0,5%. Какова вероятность того, что в партии, состоящей из 400 изделий, окажется три изделия бракованных? Решение. В условии примера дано p =0,005, n =400, m =3, следовательно, a = np =400·0,005=2. Вероятность данного события найдем по формуле Пуассона Р m , n (3,400) = 0,1804. Случайные величины и их числовые характеристики Определение 1. Случайной величиной называется переменная величина, которая в результате опыта принимает одно значение, причем неизвестно заранее, какое именно. Определение 2. Дискретной называется случайная величина, которая может принимать лишь отдельные, изолированные друг от друга значения. Случайная дискретная величина задается законом распределения, связывающим принимаемые ею значения x i и вероятности их принятия p i . Закон распределения чаще всего задается в табличной форме. Графическое представление закона распределения случайной дискретной величины – многоугольник распределения . Числовые характеристики дискретной случайной величины. 1) Математическое ожидание. Определение 3. Математическое ожидание случайной дискретной величины X с конечным числом значений называется сумма произведений возможных ее значений на их вероятности: M ( X ) = μ = x 1 p 1 + x 2 p 2 +...+ x n p n = . Вероятности всех значений случайной дискретной величины удовлетворяют условию нормировки: Свойства математического ожидания. 1 0 Математическое ожидание постоянной (неслучайной) величины С равно самой постоянной M ( C )= C . 2 0 Математическое ожидание алгебраической суммы нескольких случайных величин равно алгебраической сумме математических ожиданий слагаемых M ( X 1 ± X 2 ±...± X n ) = M ( X 1 ) ± M ( X 2 ) ±…± M ( X n ). 3 0 Константу можно вынести за знак математического ожидания M ( CX )= CM ( X ). 4 0 Математическое ожидание произведения нескольких независимых случайных величин равно произведению математических ожиданий этих величин: M ( X 1 X 2 ... X n ) = M ( X 1 ) M ( X 2 )... M ( X ) n . 2) Дисперсия дискретной случайной величины. Определение 4. Дисперсией случайной дискретной величины X называется математическое ожидание квадрата отклонения этой величины от ее математического ожидания. D ( X ) = M {[ X - M ( X )] 2 } = , где M ( X ) = μ Для вычисления дисперсии более удобна формула: D ( X )= M ( X 2 )-[ M ( X )] 2 , т.е. дисперсия случайной величины равна разности между математическим ожиданием квадрата этой величины и квадратом ее математического ожидания. Свойства дисперсии. 1 0 Дисперсия постоянной величины равна нулю D (С) = 0. 2 0 Постоянный множитель можно выносить за знак дисперсии, предварительно возведя его в квадрат: D ( CX ) = C 2 D ( X ). 3 0 Дисперсия суммы нескольких независимых случайных величин равна сумме дисперсий этих величин: D ( X 1 +...+ X n ) = D ( X 1 )+...+ D ( X n ). 4 0 Дисперсия разности двух независимых случайных величин равна сумме дисперсий этих величин D ( X - Y )= D ( X )+ D ( Y ). 3). Среднее квадратическое отклонение Определение 5 . Средним квадратическим отклонением случайной величины называется квадратный корень из дисперсии σ ( X )=. Пример. Найти математическое ожидание и дисперсию случайной величины X , которая задана следующим законом распределения: Решение. Найдем математическое ожидание: M ( x )=1·0,3+2·0,5+5·0,2=2,3. Найдем все возможные значения квадрата отклонения. [ x 1 - M ( x )] 2 =(1-2,3) 2 =1,69 [ x 2 - M ( x )] 2 =(2-2,3) 2 =0,09 [ x 3 - M ( x )] 2 =(5-2,3) 2 =7,29 Напишем закон распределения квадрата отклонения Найдем дисперсию: D ( x )=1,69·0,3+0,09·0,5+7,29·0,2=2,01. Числовые характеристики непрерывной случайной величины. Определение 6. Непрерывной называют случайную величину, которая может принимать все значения из некоторого конечного или бесконечного промежутка. Определение 7. Интегральной функцией распределения называют функцию F ( x ), определяющую для каждого значения x вероятность того, что случайная величина X примет значение меньше x , т.е. F ( x )= P ( X < x ). Свойства интегральной функции распределения 1 0 Значения интегральной функции распределения принадлежат отрезку 0≤ F ( x ) ≤1. 2 0 Функция распределения есть неубывающая функция. Следствие 1. Вероятность того, что случайная величина X попадет в интервал ( a , b ), равна приращению ее интегральной функции распределения на этом интервале P ( a < x < b )= F ( b )- F ( a ). Следствие 2. Вероятность того, что случайная непрерывная величина X примет одно определенное значение равна нулю P ( X = x 1 )=0. 3 0 Если возможные значения случайной величины X принадлежат интервалу ( a , b ), то F ( x )=0 при x ≤ a и F ( x )=1 при x ≥ a . Определение 8. Дифференциальной функцией распределения f ( x ) (или плотностью вероятности) называется производная от интегральной функции f ( x )= F "( x ). Интегральная функция является первообразной для дифференциальной функции, поэтому вероятность того, что случайная непрерывная величина x примет значение, принадлежащее интервалу ( a , b ), определяется равенством: P ( a < x < b )== F ( b )- F ( a )Зная дифференциальную функцию, можно найти функцию распределения: F ( x )= Свойства дифференциальной функции распределения 1 0 Дифференциальная функция распределения есть функция неотрицательная f ( x ) ≥0 2 0 Несобственный интеграл от дифференциальной функции распределения равен единице (условие нормировки): . 1) Математическое ожидание. Математическим ожиданием случайной непрерывной величины X , возможные значения которой прина д лежат отрезку ( a , b ), называется опр е деленный интеграл: M ( X ) = , где f ( x )-плотность вероятности случайной величины X . 2) Дисперсия. Дисперсия непрерывной случайной величины X есть математическое ожидание квадрата отклонения зтой величины от ее математического жидания D(X) = M{ 2 }.Следовательно, если возможные значения случайной величины X принадлежат отрезку ( a ; b ), то D ( x )= или D ( x )= 3) Среднее квадратическое отклонение определяется так: σ ( x ) = Пример. Найти дисперсию случайной величины X , заданной интегральной функцией F ( x )= Решение. Найдем дифференциальную функцию: f ( x )= F ’ ( x )= Выислим математическое ожидание M ( x ) = . Найдем искомую дисперсию D ( x ) = = = 2/4=4/3. Вероятность попадания нормально распределенной случайной величины X в заданный интервал Определение 9. Распределение вероятностей случайной непрерывной величины X называется нормальным, если плотность вероятности описывается формулой: , где μ - математическое ожидание, σ - среднее квадратическое отклонение. Определение 10. Нормальное распределение с параметрами μ = 0, σ = 1 называется нормированным или стандартным. Плотность вероятности нормированного нормального распределения описывается следующей формулой: . Значения данной функции для неотрицательных значений затабулированы. В силу четности функции φ ( x ) значения для отрицательных чисел легко определить φ (- x )= φ ( x ). Пример. Математическое ожидание нормального распределенной случайной величины X равно μ =3 и среднее квадратическое отклонение σ =2. Написать дифференциальную функцию X . Решение. f ( x )= Если случайная величина X распределена по нормальному закону, то вероятность ее попадания в интервал ( a , b ) определяется следующим о б разом: P(aElements of the theory of authenticity.
Basic concepts of combinatorics. The task in which it is possible to add up various combinations from the final number of elements and carry out a calculation of the number of all possible such combinations is called combinatorial.
This branch of mathematics is widely known to be practically based on the richness of natural science and technology.
Placement. Let us not take revenge n elements. Your skin is in order, so you can take revenge on m elements, called placements h n elements by m elements.
The value shows what and what is placed with n elements by m– tse m-elemental subsets that are subdivided by the composition of elements or the order of their directness.
Number of placements n elements by m The elements in the skin are identified and calculated using a formula.
Number of placements n elements by m There are only one element in the skin m successively decreasing natural numbers, of which there are more n.
For multiplicity of the creation of the first n It is customary to designate natural numbers ( n-factorial): 
Add the formula for the number of placements n elements by m elements can be written in another way: 
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butt 1. In how many ways can a group of 25 students select a group asset from the Old Age Warehouse, the Old Age Advocate, and the Trade Union Forum?
Decision. The warehouse of the group asset is an ordered multiply of 25 elements of three elements. To mean. The required number of methods is equal to the number of placement of 25 elements, three elements per skin: , or .
butt 2. Before graduation, a group of 30 students exchanged photographs. How much of everything was distributed by photography?
Decision. The transfer of a photograph from one student to another is arranged in 30 elements of two elements. The number of photographs, it seems, is equal to the number of placement of 30 elements, two elements per skin: 
.
Rearrangements. Accommodation n elements by n elements are called permutations h n elements.
The meaning means that the permutations are defined as placement. So how does skin rearrangement take place? n elements are impersonal, then various permutations are divided into one type only by the order of the elements.
Number of permutations 3 n elements of a given multiplicity are designated and calculated using the formula 
butt 3. How many four-digit numbers can be combined from the numbers 1, 2, 3, 4 without repeating?
Decision. Behind the sink there are a number of elements that need to be placed in order. Therefore, it is necessary to know the number of permutations from several elements: 
, then. From the numbers 1. 2, 3, 4 you can fold 24 numbers of four digits (without repeating numbers)
butt 4. How long can 10 guests be seated in ten places at the Christmas table?
Decision. The number of ways that can be found is equal to the number of permutations of ten elements: 
.
Poєdnannya. Let go of the impersonality that develops from n elements. The skin is its subset, which is composed of m elements, called podnannyam h n elements by m elements.
In this manner, conniving with n elements by m elements - that's all m-Elemental subsets n-elemental multiplicities, and different multiplicities are used only for those who represent a different warehouse of elements.
Multiplicities that are divided into the same type by the order of directness of the elements are not considered different.
Number of submultipliers by m elements in the skin that occur in large numbers n elements, then. number of days n elements by m elements in the skin are designated and calculated using the formula: 
or else 
.
The number has such power: 
(
).
Butt 5. How many years can it take to hold 20 football teams in a one-round championship?
Decision. Bo gra be-any team A with a team B gets together with the team B with a team A, then the skin game is combined with 20 elements of 2 each. When the number of all games is equal to the number, it is combined with 20 elements of 2 elements each: 
.
Butt 6. In how many ways can 12 individuals be divided into teams, whereas a skin team has 6 individuals?
Decision. The warehouse of the skin team consists of 12 elements of 6 each, which means that the number of methods that can be found is equal to the number of 12 elements of 6 in each: 
.
Vypadkovі podії Іmovіrіstії podії. The theory of regularities is a mathematical science that studies the patterns of phenomena. The basic understanding of the theory of authenticity lies in testing and testing.
Pid tried and tested (to be sure) understand the implementation of this complex of minds, the legacy of which will always be the same.
For example, tossing a coin is a test; the appearance of the coat of arms of the mule and the numbers – podіi.
Vipadkova's voice This is called an idea that is associated with these tests, as during these tests the tests may or may not work. The word "vypadkove" for stylost is often omitted and simply said "podiya". For example, a shot on the mark means a hit or a miss.
The idea in these minds is called reliable, if, as a result of the revelation, there is no doubt that you will wake up, and inflexible as if it obviously won’t happen. For example, a score of more than six points per hour of throwing one dice is a reliable guess; A score of ten points when throwing one dice is impossible.
Pods are called crazy Because every one of them cannot show up at once. For example, a wastage and a miss with one shot are nonsense.
It seems that there is a lot of hope in whose mind they are confirming new system I hope that as a result of this, one of them will inevitably become the result. For example, when you throw a gral brush, the points that occur when you get one, two, three, four, five or six points create another group of points.
Pods are called equal-hearted If any of them is not objectively more possible, then others. For example, at the hour of tossing the coin, the coat of arms and numbers are equally likely.
The skin has a certain level of capacity. The numerical approach to the stage of objective possibility of the idea is the validity of the idea. The popularity of the subject A signified P(A).
Get out of the system n unreasonable equal testing results m results are stored under A. Todi internationality podіi A call it a marriage m number of inheritances that belong to the A, to all the results of this testing: 
.
This formula is called the classic value of strength.
Yakshcho B- The idea is reliable, then n=mі P(B)=1; yakscho Z- the idea is impossible, then m=0і P(C)=0; yakscho A- Vipadkova podia, then 
і 
.
Thus, the trustworthiness of this position lies in the following boundaries: 
.
Butt 7. Throw the gral brush once. Find your credibility: A- Appearance of a paired number of points; B- Appearing no less than five points; C- Appeared no more than five points.
Decision. There are six equally possible independent results (the appearance of one, two, three, four, five and six points), which re-establish the system.
Podii A take three results (vipadannya two, chotirioh and six points), then 
; podіi B– two results (vipadannya five and six points), then 
; podіi C- five results (one, two, three, four, five points), then 
.
In the hour of calculating the validity, it is often necessary to use combinatorics formulas.
Let's take a look at the butts of the inexhaustible calculation of realities.
Butt 8. There are 7 red coins in the urn and 6 blue coins. Two bags can be taken out of the urns at the same time. How trustworthy is that which offends the kuli chervoni (poya A)?
Decision. The number of equal independent results is equal to 
.
Podii A hide 
results. Otje, 
.
Butt 9. A batch of 24 parts has five defective parts. In a batch, choose 6 parts. Find out whether there are 2 defective parts among 6 parts B)?
Decision. The number of equal independent results is equal.
We appreciate the variety of results m what to hide under B. Among the six parts taken from us, there are 2 defective and 4 standard ones. Two defective parts in five can be selected 
ways, and 4 standard parts from 19 standard parts can be selected 
methods.
A leather combination of defective parts can be combined with a leather combination of standard parts, that is. Otje, 
.
butt 10. Nine different books are placed in front of one police station. Know the truth of the fact that the songs of the book appear to be assigned orders Z)?
Decision. Here the number of equal independent inheritors is 
. We appreciate the variety of results T what to hide under Z. It is clear that all the songs in the book are tied together, and the tie can be separated by the police 
methods (knitting plus five other books). In the middle of the bundle, several books can be rearranged 
methods. In whom the skin combination in the middle of the ligament can join with the skin from the methods of creating the ligament, then. 
. Otje, 
.
The theory of variables and mathematical statistics
1. THEORETICAL PART
1 Sequences of fallout values and global divisions
Theoretically, it is possible to provide mothers with different types of fallout values. Let's take a look at the following main types of diversity: behind the sameness, with the sameness one, in the middle order p, behind the division.
Let go, ... - Vypadkov’s values, set in a certain worldly space (, F, P).
Meaning 1. The sequence of variable values, ... is called the magnitude, which descends to the variable value (value:), for any > 0
Meaning 2. The sequence of variable values, ... is called similar to the uniqueness of the unit (maybe singly, maybe everywhere) to the variable value, as
tobto. If there are no inheritances for which () do not converge to (), there is zero validity.
This type of familiarity is designated as follows: , either, or.
Value 3. The sequence of phased values, ... is called similar to the average order p, 0< p < , если
Value 4. The sequence of phased values is called similar to the division up to the phased value (value:), as for any interconnected non-interruptible function
The similarity in the division of the type values is determined by the terms of similarity and their function of the division. Therefore, this type of familiarity can be sensed if the variable values are set in different international spaces.
Theorem 1.
a) In order for (R-p.n.), it is necessary and sufficient for anyone to > 0
) Sequence () is fundamental with respect to one thing or another, if for any > 0.
Finished.
a) Let A = (: |- | ), A = A. Todi
Therefore, the hardening of a) is the result of an offensive lanyard with the implication:
P(: )= 0 P() = 0 = 0 P(A) = 0, m 1 P(A) = 0, > 0 P() 0, n 0, > 0 P( ) 0,
n 0, > 0.) Significantly = (:), = . Then (: (()) is not fundamental) = and so it is, as in a) it appears that (: (()) is not fundamental) = 0 P() 0, n.
The theorem has been proven
Theorem 2. (Cauchy Criterion may even sing)
In order for the consistency of the variable values () to be similar to the homovirality of one (up to a certain variable value), it is necessary and sufficient that there be a fundamental one with the homovirality of one.
Finished.
Yakshcho, then +
It is clear that there is a need for mental theorems.
Let us now sequence () is a fundamentally important unit. Significantly L = (: (()) is not fundamental). Then for all numerical sequences () is fundamental and, according to the Cauchy criterion for numerical sequences, fundamental (). We agree
Thus, the singing function has a variable value i.
The theorem has been proven.
2 Method of characteristic functions
The method of characteristic functions is one of the main features of the analytical apparatus of the theory of validity. In order to use variable values (which take actionable values), the theory of characteristic functions will require obtaining complex-valued variable values.
There is a lot of significance and power, which is similar to the magnitude of the fall, and is easy to tolerate and complex fall. So, mathematically, M ?complex-valued variable value ?=?+?? is respected by the song, as indicated by the mathematical calculation of M ?there ?. Whose decision is important to M ?= M ? + ?M ?. Because of the importance of the independence of the fall elements, it becomes clear that the values are complex ?1 =?1+??1 , ?2=?2+??2independent todi and even more so, if independent pairs of variable values ( ?1 , ?1) that ( ?2 , ?2), or else they are independent ?-algebra F ?1, ?1 and F ?2, ?2.
Order from space L 2of the active phased quantities with the final other moment it is possible to introduce the Hilbertian space of complex-valued phased quantities ?=?+?? z M | ?|2?|2= ?2+?2, that scalar creation ( ?1 , ?2)= M ?1?2¯ , de ?2¯ - The Vipadkova value was obtained in a complex manner.
During algebraic operations, vectors Rn are considered as algebraic elements,
As a row vector, a * - (a1, a2, ..., an). If Rn, then their scalar value (a,b) will be understood. I realized that
Yakshto aRn ta R=||rij|| is a matrix of order nхn, then
Value 1. Let F = F(x1,….,xn) - n-virtual function of the division (, ()). Its characteristic function is called the function
Vicennia 2 . What? = (?1,…,?n) - a phase vector of values in the global space with values, whose characteristic function is called a function
de F? = F?(x1,….,xn) - function of the subdivision of the vector?=(?1,…,?n).
Since the function of the division F(x) equals f = f(x), then
And here the characteristic function is nothing more than a transformation of the Fourth function f(x).
From (3) it follows that the characteristic function ??(t) of the fall vector can also be calculated
Main characteristics of characteristic functions (for times n=1).
Let's go? =? (?) - Vipadkova value, F? =F? (x) - this is a subdivision function and is a characteristic function.
Slide signify, yakscho, then.
True,
where we have discovered that the mathematical calculation of the creation of independent (boundary) variable quantities is older than the creation of their mathematical calculation.
Power (6) is the key to confirming boundary theorems for sums of independent variable quantities using the method of characteristic functions. In this connection, the function of the division is expressed through the functions of the division of the adjacent subdivisions in a clearly folded order, the sign * itself means a handful of divisions.
The cutaneous function of the subsection can be associated with the drop value, which is the function of this function within its function of the subsection. Therefore, for the presentation of the power of characteristic functions, one can look at the characteristic functions of the variable quantities.
Theorem 1. Let's go? - Vipadkova value with the division function F = F (x) і - її characteristic function.
The following characteristics are present:
) Continuously without interruption;
) has an effective function even if the division F is symmetrical
) what for active n? 1, then in all likelihood there will be
)As it starts and ends, then
) Say hello to everyone n ? 1 ta
for everyone |t| The next theorem shows that the characteristic function uniquely signifies the division function. Theorem 2 (unity). Let F and G be two functions of a subdivision that have one and the same characteristic function, so that for all The theorem shows that the division function F = F(x) is uniquely consistent with its characteristic function. The next theorem gives explicit expression of the function F through. Theorem 3 (Uzagalnenny formula). Let F = F(x) be the subdivision function and the characteristic function. a) For any two points a, b (a< b), где функция F = F(х) непрерывна, ) Since the function of the division F(x) is the thickness f(x), Theorem 4. In order for the components of the fall vector to be independent, it is necessary and sufficient that its characteristic function be the creation of the characteristic functions of the components: Bochner-Khinchin theorem .
Let it be an uninterrupted function. In order for it to be characteristic, it is necessary and sufficient for it to be invisible and significant, so that for any active t1, …, tn and any complex numbers Theorem 5. Nehai is a characteristic function of the fall value. a) If for a person, then the vipadkova value is often with the scale, then ) If there are two different points, which is an irrational number, then is it a different value? є virogen: de a – deyaka constant. c) What is the Vipadkova value? virogena. 1.3 Central boundary theorem for independent but subdivided variable quantities Nehai () - Sequence of independent, however, subdivided phased quantities. Mathematical calculation M= a, dispersion D= , S = , and Ф(х) is a subdivision function of the normal law with parameters (0,1). Let us also introduce the sequence of fall values Theorem. Yakshcho 0<<, то при n P(< x) Ф(х) равномерно относительно х (). In this case the sequence () is called asymptotically normal. Because M = 1 and from the continuity theorems, it follows that in order of weak convergence, the FM f() Mf() for any continuous interconnected f may also converge M f() Mf() for any continuous ї f , such that |f(x)|< c(1+|x|) при каком-нибудь. Finished. Evenly boiled with the same low boiling and uninterrupted f(x). Further, without limiting the complexity, we must use a = 0, since otherwise it would be possible to look at the sequence (), but our sequence () would not change. Now, to prove the necessary simplicity, it is necessary to show that (t) e, if a = 0. We can (t) = de = (t). So as it is M, then it is fair and fair Ozhe, for n The theorem has been proven. 1.4 Basic tasks of mathematical statistics Their short description The establishment of patterns that order mass phenomena is based on statistical data - the results of caution. The first department of mathematical statistics is to indicate the methods of collecting and grouping statistical data. Another task of mathematical statistics is to develop methods for analyzing statistical data for research purposes. In the case of any mathematical statistics department, there are two pieces of information in its disposal. The first and most important (obvious) is the result of observation (experiment) as a selection from any general population of scalar or vector variable quantities. In this case, the selections may be fixed, and may increase during the course of the experiment (then the subsequent procedures of statistical analysis may be affected). Another thing is that this is all a priori information about the power of the object that is being accumulated up to the current moment. Formally, a priori information is displayed in the output statistical model that is chosen for the most important task. However, it is not possible to talk about the closeness of the primary sense of trust based on the results of the investigations. When approaching significant figures of any size, ask for respect that you can order between death, even if there is no mercy. The frequency of occurrence is similar to any number of traces through the variability of the results of several traces. Due to the variability of the results of some subsequent studies, the frequency can significantly differ from the reliability of the field. Therefore, due to the unknown reliability of this phenomenon and the frequency of this phenomenon for a large number of traces, we cannot indicate between the abductions and guarantee that the abduction will not occur between them. Therefore, in mathematical statistics, we should talk not about the approximation of the values of unknown quantities, but about their relative values, estimates. Predetermined estimation of unknown parameters occurs in cases where the function of the subdivision of the population is given with precision to the parameter. In which case it is necessary to know such statistics, the sample values for the analyzed implementation of xn sample samples can be taken into account with nearby parameter values. Statistics, sample values of any implementation xn are taken to be close to the value of an unknown parameter, called a point estimate and assigned an estimate, and - to the values of a point estimate. The point estimate can be satisfied entirely with the benefit of ensuring that the selected value corresponds to the reference value of the parameter. Is it possible to use another approach to complete the task: to find such statistics in order to be more reliable? nervousness came to an end: Why talk about interval assessment? Interval called a confidence interval with a confidence coefficient? Having evaluated the results of this study and another statistical characteristic, the question arises: to what extent is it acceptable to support the assumption (hypothesis) that the unknown characteristic has the same meaning as determined as a result of the evaluation? This is how another important class of problems in mathematical statistics arises – the task of testing hypotheses. The singer's task is to verify the statistical hypothesis before specifying the parameter estimate. When estimating a parameter, we know nothing about its value. When revising a statistical hypothesis, the value is transferred to the known value and it is necessary to verify the assumption based on the results of the experiment. In many works of mathematical statistics, the sequences of variable quantities are considered, which converge in one sense to another (variable values and constants), if. Thus, the main tasks of mathematical statistics are the development of methods for finding estimates and tracking the accuracy of their proximity to the estimated characteristics and the development of methods for testing hypotheses. 5 Verification of statistical hypotheses: basic concepts The task of developing rational methods for testing statistical hypotheses is one of the main tasks of mathematical statistics. A statistical hypothesis (or simply a hypothesis) is called a statement about the type or power of the subdivision of random quantities that are observed in an experiment. Do not select, which is the implementation of a phased selection from the general population, the density of the division which lies under an unknown parameter. Statistical hypotheses where the true value of a parameter is unknown are called parametric hypotheses. If something is a scalar, then we are talking about one-parameter hypotheses, and if it is a vector, then we are talking about multiparametric hypotheses. A statistical hypothesis is called a simple hypothesis, as it appears de - The value of the parameter is specified. The statistical hypothesis is called foldable, as it appears de - a number of parameter values that add up to more than one element. Once we recheck two simple statistical hypotheses, we see de - two tasks (different) values of the parameter, one hypothesis is called the main one, and the other one is called an alternative or competing hypothesis. A criterion, or a statistical criterion, for testing hypotheses is the rule by which, based on sampling data, decisions are made about the validity of either the first or another hypothesis. The criterion is set using an additional critical multiplier, which is a submultiple of the sample space of the fall sample. The decision is taken like this: ) if the selection falls on a critical multiplicity, then the main hypothesis is given and an alternative hypothesis is accepted; If the sample does not correspond to a critical multiplicity (an additional multiplicity must be added to the sample space), then an alternative hypothesis is raised and the main hypothesis is accepted. If you choose any criterion for the possible removal of the following types: 1) accept the hypothesis, if it is correct - mercy for the first generation; )accept the hypothesis, if it is true, it is a mercy of a different kind. The certainty of the execution of mercy to the first and the other kind is indicated by: there is confidence in the opinion that the hypothesis is valid The consistency of mercy of the first kind is called an equal criterion of significance. The value that is sufficiently reliable to establish the main hypothesis, if it is true, is called the strength criterion. 1.6 Independence criterion Є selection ((XY), …, (XY)) from a two-dimensional division L with an unknown function of the division, for which it is necessary to verify hypothesis H: for some one-dimensional functions of the division. A simple criterion for hypothesis H can be obtained based on the methodology. This method is used for discrete models with a terminal number of results, so it is important to note that the variable value increases the terminal number s of the values that are significant by the letters, and the other component - the k value. Since the output model has a different structure, the possible values of the variable values are first grouped along the first and other components. And here the impersonal value is divided into s intervals, the impersonal value - into k intervals, and the impersonal value itself - into N=sk straight cutters. It is significant through the number of bets (the number of elements of the selection that lie in the rectangular one, as the data are grouped), then. The results can be carefully analyzed manually as a table of conjugation of two characters (Table 1.1). In the additions and the name mean two signs that are used to classify the results of precaution. Let R, i = 1, ..., s, j = 1, ..., k. So the independence hypothesis means that there are s+k stationary ones like i. Table 1.1 Suma . . .. . .. . . . . .. . .. . . . . . . . . . . . . . .Suma . . .n Thus, the hypothesis H is reduced to the affirmation that the frequencies (the number of their relatives N = sk) are distributed according to the polynomial law with the properties of the inheritances, so that a specific structure is assigned (the vector of the properties of the inheritances in is indicated by the values of r=s+k-2 unknown parameters. To verify this hypothesis, we find maximum likelihood estimates for the initial scheme of unknown parameters. If the null hypothesis is true, then the likelihood function looks like L(p) = where the multiplier of unknown parameters does not lie. Based on Lagrange's method of non-significant multipliers, we can see that the estimates that are being looked for, seem to be Ozhe, statistics L() when, as a result, the number of steps of freedom in the boundary section increases to N-1-r=sk-1-(s+k-2)=(s-1)(k-1). However, for the sake of the great ones, one can use the following rule for verifying a hypothesis: the hypothesis N is given either or not, if the calculation of the actual data of the statistical value satisfies the inequalities This criterion may asymptotically (when) assign a level of significance and is called the criterion of independence. 2. PRACTICAL PART 1 Connection of problems with types of profitability 1. To convey that the dangers of the world may be heard in a sing-song manner. Point the control butt, which shows that the turnaround is incorrect. Decision. Let the sequence of fall values descend to the fall value x as melodiously as possible. So, for someone? > 0 So that And from the shortness of xn to x, it sings in a melodious way that xn will go to x for its cross-borderness, which is why Ale zvorotne firmeniya is incorrect. Let's say - the sequence of independent phased values that represent the same subdivision function F(x), which is equal to zero at x? 0 and equal to x > 0. Let’s look at the sequence This sequence goes to zero due to its homogeneity, fragments pragne to zero for whatever is fixed? V. Prote reduction to zero may not be melodious. Actionable Pragne to one, then with the reliability of 1 for any and n in sequence there will be implementations that will be reversed?. It is significant that due to the obviousness of many additional minds that are superimposed on the values of xn, the increase is incredibly strong and even melodious. Let xn – monotonous sequence. Bring to the point that the reduction of xn to x is incredibly gravitational towards the reduction of xn to x with the reduction of 1. Decision. Let xn - monotonously descending sequence, then. To simplify our understanding, it is important that x º 0, xn ³ 0 for all n. Don’t let xn go to x for your confidence, there may not be a sing-song voice about it. Are you still asleep? > 0, so that with usikh n Ale y said means that for all n It’s important to note the compatibility of xn to x for its internationality. In this way, for a monotonous sequence xn, which goes up to x for homovirality, there is a place and convergence with homogeneity 1 (maybe melodiously). Please let the sequence of xn go to x for internationality. Bring that from this sequence you can see the sequence that comes down to x with the same level of 1 at. Decision. Let there be a sequence of positive numbers, and i are such positive numbers that they are a series. We will determine the sequence of indexes n1 Todi row So if the series converges, then what? > 0 surplus in the row to zero. Ale todi pragne zero i Let us know that in the middle, in any positive order, in a positive way, there is a difference between mutuality and homovirality. Point the butt, which shows that the turnaround is incorrect. Decision. Let the sequence xn descend to the value x y of the middle order p > 0, then Chebishev's accelerated unevenness: for the happy? > 0 i p > 0 Having straightened it out to the doctors, we reject it, we then xn go to x for international assistance. However, the increase does not necessarily lead to the increase in the middle order p > 0. This shows the attacking butt. Let's take a look at the world space áW, F, Rñ, where F = B is the Borel s-algebra, R is the Lebesgue world. What is significant is the sequence of the fall values in the following order: Sequence xn goes to 0 for consistency, fragments but for whatever reason p > 0 So the average person doesn’t have a life. Nevermind what everyone cares about. Bring that in this range xn goes to x y mean square. Decision. Respectfully, te y. Let's reject the rating. Let's take a look at the Vipadkova value. Let's go? - Quite a positive number. Todi pri ta for. Yakshcho, then y. Otje, . What about the fragments? Since i is always small, then at, then in the mean square. Bringing xn to converge to x is incredibly easy, which means there is little danger. Point the control butt, which shows that the turnaround is incorrect. Decision. It is clear that at the skin point x, which is the point of continuity (which is necessary and sufficient for a weak mind), is a function of the subdivision of the value xn, and - the value of x. Let x be the point of non-interruption of function F. If so, then we rightly accept one of the inequalities either. Todi Similarly, if you just want one of the inequalities or Yes, then for how many small things? > 0 it is also N, so with all n > N On the other hand, since x is a point of continuity, can we find out this way? > 0, which is for how small a year So, for how many little ones every year? And it is also N, which for n >N or else, those same This means that all points of continuity have the same place. So, because of the sufficiency, the weakness of the sufficiency is incredibly strong. The resoluteness, which seems to have burned out, has no place in going back. To get to the bottom of this, let us take the sequence of variable values that are not equal to the magnitude of 1 constant ones and represent the same function to the subdivision F(x). It is important that for all purposes the values are independent. Obviously, the weakness is due to the fact that all members have the same function in the subdivision. Let's take a look: |Due to the independence and the new dispersion of values, it appears that Let us choose among all the functions of subdivisions of non-generated phase values such that F(x) will be equal to small ones from zero? Then there will be no difference between zero and unbound growth. 7. Let there be a weak shortening, but it is stable. Bring that in this situation they will converge to the point of completeness. Decision. Let go of the credibility of 1 ancient a. Therefore, weak friendliness means death for others. So, then when and when. That's for and for. The star is screaming, what for anyone? > 0 availability jump to zero at. Tse means that pgne to zero at, then converge to homovirality. 2.2 Connection of tasks to the central heating center The values of the gamma function Г(x) for x= are calculated by the Monte Carlo method. We know the minimum number of tests necessary to ensure that, with a homovirality of 0.95, it is possible to determine that the apparent loss will be calculated to be less than one hundredth. For accuracy up to date Vidomo Having made the substitution in (1), we arrive at the integral over the end gap: With us, that Apparently, clearly in sight, de, and divided evenly into. Let the statistical testing be completed. Another statistical analogue is the quantity de, - independent variable values with an even distribution. With this The CPT is a trace that is asymptotically normal to the parameters. This means that the minimum number of tests that will ensure a significant reduction in the cost of a little more. There is a sequence of 2000 independent but divided variable values with mathematical calculations equal to 4 and dispersion equal to 1.8. The arithmetic mean of these quantities is a linear value. Calculate the probability that the random value will be in the interval (3.94; 4.12). Let it be, ...,... - the sequence of independent fall-off values, however, the new division is M=a=4 and D==1.8. Then until the sequence () the CPT becomes stagnant. Vipadkova value The probability of what will happen in the intervals (): At n=2000, 3.94 and 4.12 are eliminated 3 Verification of hypotheses using the criterion of independence As a result of the investigation, it was established that 782 light-eyed blue fathers still have light eyes, and 89 light-eyed blue fathers have dark eyes. 50 dark-eyed fathers have blue eyes, and 79 dark-eyed fathers have blue eyes. What is the difference between the color of the father’s eyes and the color of the eyes of their blue ones? The value is accepted as equal to 0.99. Table 2.1 ChildrenFathersMamasLight-eyedDark-eyedLight-eyed78279861Dark-eyed8950139Amount8711291000 H: there is no difference between the colors of the eyes of children and fathers. H: There is a difference between the colors of the eyes of children and fathers. s=k=2 =90.6052 from the first stage of freedom The calculations were made using the Mathematica 6 program. Remnants > , then hypothesis H about the prevalence of correlation between the color of fathers’ and children’s eyes, with equal significance, should be modified and accepted as an alternative hypothesis H. It is confirmed that the result of the liquid depends on the curing method. Turn over the statements for the data listed in the table. 2.2 Rhubarb trust to accept the highest 0.95. Table 2.2 ResultMethod zastosuvannya ABCUnfriendly111716Agreeable202319 Decision. To solve this problem, use the table of the receipt of two signs. Table 2.3 ResultMethod zastosuvannyaAmount ABCUnfriendly11171644Agreeable20231962Amount314035106 H: the result of the liquids does not remain due to the curing method H: the result of the liquids depends on the curing method Statistics are calculated using this formula s=2, k=3, =0.734626 with 2 degrees of freedom. Calculations are allocated from the Mathematica 6 program According to the tables, we know that. Oskolki< , то гипотезу H, про отсутствия зависимости действия лекарств от способа применения, при уровне значимости, следует принять. Visnovok This work has made theoretical contributions from the section “Criterion of Independence”, as well as “Boundary Theorems of the Theory of Independence”, the course “Theory of Independence and Mathematical Statistics”. During the course of the war, the criterion of independence was re-verified in practice; Also, for specifying sequences of independent variable quantities, the theory of the central boundary theorem was verified. This work helped to further my knowledge of these branches of the theory of independency, work with literary elements, and firmly established the technique of revising the criterion of independence. unique statistical hypothesis theorem Perelik Posilan 1. Collection of instructions based on the theory of reliability with decoupling. Uch. Pos_bnik/Ed. V.V. Semenets. – Kharkiv: KhTURE, 2000. – 320 p. Gikhman I.I., Skorokhod O.V., Yadrenko M.I. The theory of variables and mathematical statistics. – K.: Vishcha school, 1979. – 408 p. Ivchenko G.I., Medvedev Yu.I., Mathematical statistics: Navch. Handbook for colleges. - M: Visch. school, 1984. - 248 p., . Mathematical statistics: Navch. for universities/V.B. Goryainov, I.V. Pavlov, G.M. Tsvetkova ta in; Per ed. V.S. Zarubina, A.P. Krischenko. - M: View of MDTU im. Not. Bauman, 2001. – 424 p. Do you need help from those who are vaccinated?
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