The rules and procedures of mathematical statistics spiral on. Schasne uyavlenya about mathematical statistics. Empirical function of rozpodilu, histogram

From mathematical statistics to the mind “distributed mathematics, assignments to mathematical methods for selection, systematization, processing and interpretation of statistical data, as well as registration for science or practical applications. The rules and procedures of mathematical statistics spiral on the theory of imaging, which allows the assessment of the accuracy and reliability of problems, obsessed in skin problems on the basis of explicit statistical material. " At the same time, statistical data is called a kind of information about a number of objects in any big-mensh great sukupnosty, which volodyut these signs.

According to the type of viruses, the mathematical statistics are divided into three sections: the description of the data, the assessment and the revision of hypotheses.

By the type of statistical tributes that have been smashed, mathematical statistics can be divided into straight lines:
- one-dimensional statistics (statistics of various values), in which the result of caution is described by an arbitrary number;
- Bogatovimіrny statistical analysis, de the result of caution over the object is described by decilcom numbers (vector);
- statistics of vypadkovyh processes and hour rows, de the result of caution - function;
- statistics of objects of non-numerical nature, in which the result of being careful about non-numerical nature, for example, є non-numerical ( geometric figure), The ordering of the changes in the result of the change for the clear acquaintance.

Historically, the first appeared in the field of statistics on non-numerical nature (zokrema, the establishment of an assessment of a part of the club and revisions of hypotheses about it) and one-dimensional statistics. The mathematical apparatus is simpler for them, so on their applications you want to demonstrate the basic ideas of mathematical statistics.

There are no methods of processing data, so mathematical statistics, є provable, which spiral on the imovirnіsnі models of evіvаlу real events аnd processes. Read more about models of behavior of survivors, determination of risky, function of technological possession, rejection of results in an experiment, overcoming anxiety, etc. The model of a real phenomenon follows the prompted, as it were, the magnitudes and the connection between them, in terms of the theory of imovities. The constancy of the real model of reality, that is, the adequacy, to bring up, in addition to the statistical methods of converting hypotheses.

The improbable method of processing the data is sound, it can be victorious with an earlier analysis of the data, since the stench does not give the possibility of assessing the accuracy and reliability of the data, which are taken on the basis of statistical material.

Imovirnіsnі and statistical methods of stagnation of the process. Їх zasosuvannya obov'yazkovo, if broken on the basis of vibrating gifts of visnovka are transferred to the entire sukupnіst (for example, from vibrating to the entire batch of products).

In specific areas, the definition of a vicarious method is based on both the statistical and statistical methods of a wide distribution, as well as specific. For example, in the distribution of viral management, assigned to statistical methods for managing the quality of products, I will apply mathematical statistics (including the planning of experiments). For additional methods, a statistical analysis of the accuracy and stability of technological processes and a statistical assessment of the quality is carried out. Prior to specific methods, there should be methods of statistical acceptance control of product quality, statistical regulation of technological processes, assessment and control of reliability and in.

Such applied and statistical-statistical disciplines, such as the theory of hope and the theory of mass service, are widely used. The name of the first one is clear from them, a friend to be engaged in systems such as telephone exchanges, on how at any given moment there are wikkicks - vimogi of subscribers, who dial numbers on their telephone devices. The triviality of the service staff is tsikh vimog, that is, the triviality of the roses, can also be modeled in small quantities. A great contribution to the development of cich disciplines was made by a corresponding member of the Academy of Sciences of the USSR A.Ya. Khinchin (1894-1959), Academician of the Academy of Sciences of the URSR B.V. Gnedenko (1912-1995) and іnshі vіtchіznyani vchenі.

Zm_st.

1. Introduction:
- How can the theory of imaging and mathematical statistics be vikoristic? - side. 2
- So is the "mathematical statistics"? - side. 3
2) Apply the definition of theory and mathematical statistics:
- Vibirka. - side. 4
- Zavdannya appraisal. - side 6
- Jovial-statistical methods and optimization. - side. 7
3) Visnovok.

Entry.

How can the theory of imaging and mathematical statistics be vikoristic? The discipline is the basis of the mathematical-statistical methods of taking decisions. In order to speed up with a mathematical apparatus, it is necessary to adopt the solution of visibility in terms of mathematical and statistical models. The stagnation of a specific ymovinno-statistical method takes the decision to be made up of three stages:
- transition from economic, managerial, technological reality to an abstract mathematical and statistical scheme, so that an imoving model of a control system, technological process, procedures for taking decisions, winter conditions, etc.
- Carrying out rosary and rejection of visnovka by purely mathematical methods within the framework of an imovirn_naya model;
- іnterpretatsіya mathematical and statistical character visnovkіv stosovno realnoї situatsії i of acceptance vіdpovіdnogo rіshennya (napriklad, about vіdpovіdnіst abo nevіdpovіdnіst yakostі produktsії vstanovlenim vimogam, neobhіdnostі nalagodzhennya tehnologіchnogo etc. Process i) zokrema, ukladennya (about chastku defective odinits produktsії in partії, specifically about type of laws for the development of controlled parameters in the technological process and in.).

Mathematical statistics of the vicarist understanding, the methods and results of the theory of immorality. The main food is easy to understand and motivate the smart models to accept solutions in economical, managerial, technological and other situations. For active and correct registration of normative-technical and instructive-methodical documents on the basis of statistical and statistical methods, the solution is required in advance of knowledge. So, it is necessary for the nobility, for which minds it is necessary to collect that document, as it is necessary for the mother to receive the information for its choice, and as the decision to be guilty is taken for the results of the processing of the data, etc.

Why is statistics also "mathematical"? From mathematical statistics to the mind “distributed mathematics, assignments to mathematical methods for selection, systematization, processing and interpretation of statistical data, as well as registration for science or practical applications. The rules and procedures of mathematical statistics are spiraling to the theory of imaging, which allows the assessment of the accuracy and reliability of problems, obsessed in skin problems on the basis of explicit statistical material. " At the same time, statistical data is called a kind of information about the number of objects in any big-mensh great sukupnosty, which volodyut these signs.

According to the type of viruses, the mathematical statistics are divided into three sections: the description of the data, the assessment and the revision of hypotheses.

By the type of statistical tributes that have been smashed, mathematical statistics can be divided into straight lines:

The statistics are of the same type (statistics of various values), in which the result of caution is described by an arbitrary number;

Bagatovimirny statistical analysis, de the result of caution over the object is described by decilcom numbers (vector);

Statistics of vypadkovyh processes and hour rows, de the result of the watchdog - function;

Statistics of objects of non-numerical nature, in which the result of being careful is non-numerical nature, for example, є without lichchy (geometric figure), ordering or discarding as a result of being judged for a clear familiarity.

Apply the definition of theory and mathematical statistics.
It is easy to understand the number of applications, if the probabilistic and statistical models are a good tool for the operation of management, virobnichi, economics, and national administrations. So, for example, a coin, yak vikoristovuyu yak a stallion, is guilty of being "symmetrical", so that with її kidannі in the middle, in half of the wipes, the coat of arms is guilty, and in half of wipads - the lattice (tails, number). Does Ale scho mean "in the middle"? If you spend a lot of gray 10 throws in a leather series, then often there will be a set of gray, in which a coin is 4 times a coat of arms. For a symmetrical coin, the price will be 20.5% gray. And if 40,000 emblems appear on 100,000 throws, then why can you put a symmetrical coin? The procedure for taking decisions will be based on the theory of imperatives and mathematical statistics.

Razglyanuty the butt can be lacking in seriousness. However, it is not so. Zherebkuvannya widely vikoristovuєtsya at organіzatsії Promyslova tehnіko-ekonomіchnih eksperimentіv, napriklad at obrobtsі rezultatіv vimіryuvannya pokaznika yakostі (time Tertea) pіdshipnikіv in zalezhnostі od rіznih tehnologіchnih faktorіv (vplivu lepohіna seredovischa, metodіv pіdgotovki pіdshipnikіv before vimіrom, vplivu navantazhennya pіdshipnikіv in protsesі vimіryuvannya i t. NS.). Supposedly, it is necessary to determine the quality of the bears in the amount of oil from the results of the pick-up in the old canned oils, so that in the warehouses A and B. so, you will be able to eliminate the sub-activism and prevent the activity of the adopted solution.

vibirka
You can go to the food chain to get rid of the help of the foal. A similar butt can be brought with the control of the quality of any product. If you want to see it, it’s not like it’s going to be controlled by the batch of products installed by the vimogs, you’ll see a vibe from it. For the results of the control of the vibrating machine, there is a scrutiny about the whole party. In general, it is even more important to eliminate sub-activism in the form of vibrations, so it is necessary that the skin unit of products in the control of the party is small, however, the quality of being seen in vibrations. In the wicked minds, there is a single product, and in the wobble, you should be invited to come here not for the help of the foal, but for the special tables of the special numbers, or for the help of the computer sensors of the numbers.
Analogous problems of securing the activity of the decision are made when new schemes for organizing the business are introduced, payment for prats, when tenders and competitions are held, candidates are selected for vacancies, etc. Everywhere a foal is required or a separate procedure is required. It can be explained on the application that the most powerful and other force behind the command's power when organizing the tour for the Olympic system (xto having progressed vibuvay). Don't know if the team is strong enough to rewrite the strongest team. It is clear that the best team will definitely become the champion. A friend for the strength of the team will go to the final tody and only to the final, if until the final she will not have Igor with the Maybutnim champion. If it’s planned, then a friend for the strength of the team at the finals will not indulge. That, hto planu tournіr, maybe even enough to "vibrate" a friend for the strength of the team from the tournіr, calling in the first operation with the leader, or taking care of the other team, failing the team with the weak ones. Schob uniknuti sub'activism, carry out the colt. For a tour of 8 teams, it is important to have two strongest teams in the finals, the way is 4/7. According to the friend's imovirnistu, the team will leave the tour without delay.
Whenever there is a single product (with the aid of a caliper, micrometer, ammeter, etc.), mistakes are made. For a systematic abduction, it is necessary to adapt the bagatorazovannya of one product, the characteristics of one type (for example, a standard form). With a lot of memory, there is a lot of systematic abduction of the presence and vipadkov abduction.

To that, the food is fasting, as the results of vimiryuvan diznatisya, є l systematic hacking. If it is only by definition, if є is rimmed in case of Chergovy vimir, positive or negative, then the process can be brought up to the forefront. Surely, it is possible to change the scale from the coins of the coin, positively - from the coat of arms, negatively - the lattice (zero deflection with sufficient number of scale frames is practically not used). Todi reversal of the day-to-day systematic misbehavior is equivalent to the reversal of the symmetry of the coin.

By means of the cikh mirkuvan є the establishment of the reconciliation of the day-to-day systematic fraud to the task of reconfiguring the symmetry of the coin. Carrying out the world culture is brought up to the so-called "criterion of signs" in mathematical statistics.
"Sign test" is a statistical criterion that allows you to reconsider a null hypothesis, so that the vibration is assigned to a binomial distribution with the parameter p = 1/2. The criterion of signs can be considered as a nonparametric statistical criterion for reconverting the hypothesis of the equality of the media to the given value (spring, zero), and also the visibility of the sound (the visibility of the effect of the two sounds) in It also allows us to reconsider the hypothesis of symmetry, however, for a wide range of criteria, it is Wilcoxon's one-shot criterion and its modification.

With the statistical regulation of technological processes on the basis of methods of mathematical statistics, the rules and planning of statistical control of processes are developed, directing on the timely development of technological processes and when entering before the improvement of the production and production of products. And go straight to the fast vitrat vyrobniztva and go to see the production of obscure products. With statistical acceptance control on the basis of methods of mathematical statistics, plans are developed to control the quality by way of analysis of vibrations from product lots. The versatility of the field lies in the fact that it is correct to be able to use the correct statistical models for taking decisions, on the basis of which it is possible to take into account the food supply. Mathematical statistics for a complete breakdown of imovirnіsnі models and methods of reconversion of hypotheses, sprays, hypotheses about those, where a part of defective product units is of the appropriate number p0, for example, p0 = 0.23.

Zavdannya appraisal.
In a number of administrative, financial, economic, national and cultural situations, there is an established type - the establishment of an assessment of the characteristics and parameters of the parameters of the parameters.

The butt is visible. Let the party take control of the N electric lamps. With the tsієї party in vypadkovy rank vіdіbrana vibіrka obsyag n electric lamps. Vinikaє a number of natural foods. As for the results of the viprobation of vibrating elements, what is the significance of the average term of the service of electric lamps, and how accurate is it possible to evaluate the characteristic? How can you change the accuracy, how can you take the vibrator of a big one? For every number of years T, is it possible to guarantee that not less than 90% of electric lamps will last T and more years?

Admittedly, when vibrating vibrators with a lot of n electric bulbs, X electric bulbs are defective. Today the food is coming. How many boundaries can be used for the number D of defective electric lamps in a batch, for equal D / N defectiveness, etc.?

For with a statistical analysis of the accuracy and stability of technological processes, such indicators of quality, such as the average value of the monitored parameter and the steps of this process, are assessed. From the standpoint of the theory of imperfections, the mean value of this value is not mathematically accurate, and in terms of the statistical characteristics, the variance, the mean-square value of the variance. How to evaluate the statistical characteristics of the vibrating data and how accurate the price is? Similar stocks can be adjusted even more. It is important to show here how the theory of imaging and mathematical statistics can be used by the general management in the process of taking decisions in the field of statistical management of product quality.

Ymovinno-statistical methods and optimization. The idea of ​​optimization permeates the currently applied mathematical statistics and statistical methods. And itself, the methods of planning experiments, statistical acceptance control, statistical regulation of technological processes and in. On the other hand, optimizing the setting in the theory of acceptance of solutions, for example, the applied theory of optimizing the quality of products and standards, transferring a wide range of mathematical and statistical methods, first for all applied statistics.

In the virobny management, during the optimization of the quality of the product and in the production of standards, it is especially important to establish statistical methods on the cob stage of the life cycle of products, so that on the stage of the scientific and design box). It is necessary to explain the interconnection of information available on the cob stage of the life cycle of the product, and the need to predict the technical capabilities and economic situation in the future. Statistical methods of blame for the determination of optimization at all stages - when scaling changes, development of mathematical models of functioning of viruses and systems, carried out by technical and economic experiments, etc.

At the staff of optimization, including the optimization of product quality and standards, all areas of statistics are victorious. And itself, the statistics of large quantities, a rich statistical analysis, statistics of large processes and hour series, statistics of objects of non-numerical nature. Vibir of the statistical method for the analysis of specific donations is carried out according to the recommendations.

Visnovok.
V
etc .................

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Mathematical statistics- a lot of fun in mathematical science, how to deal with the statistical description of the results of experiments and by the incentive mathematical models to take revenge on the understanding imovirnostі. To serve as a theoretical base of mathematical statistics theory of imovirnost.

The structures of mathematical statistics traditionally have two main sections: opisova statistics and statistical visnovka (Fig. 1.1).

Small. 1.1. Basic sections of mathematical statistics

opisova statistics to be victorious for:

o publicity indicators of one change (statistics of the last vibration);

o Determination of interconnection between two houses and more changes (correlation-regression analysis).

The descriptive statistics give the ability to revise new information, better intelligence and generally evaluate it, so that I will describe the scientific function of the information, which I will give my name. Methods of descriptive statistics for the click to reconfigure the definition of empirical tributes to the system of specific forms and numbers for the reception: frequency distribution; indicators of tendencies, variability, connection. With the help of methods, we can use the methods to develop statistics in the form of vibration, which serve as a guide for the establishment of statistical data.

Statistical Visnovki give you the power:

o assess the accuracy, reliability and efficiency of vibrating statistics, know the pitfalls that occur in the process of statistical data (statistical evaluation)

o publicity of the parameters of the general sukupnosti, rejected on the display of vibrating statistics (revision of statistical hypotheses).

Head meta scientific doslіdzhen- the idea of ​​rejecting the new knowledge about the great class of manifestations, especially for the reason, which is accepted as the name of the general sukupnost.

General Sukupnist- the price of the supremacy of the objects for the release, vibirka- її part, the yaka is molded in a singing science-based way 2.

The term "general sukupnist" vikoristovuyutsya todі, since mova ide about the great, ale kintsevu sukupnіst of pre-sludzhuvanih ob'octіv. For example, about the sukupnosti abіturієntіv Ukraine in 2009 rotsi abo sukupnіst children preschool education Mista Rivne. General sukupnosti can reach significant commitments, but they are kintsevym and indefinite. In practice, as a rule, I can pay the right with Kintsev sums. If it is necessary to pay a general fee to the amount of 100, then, according to the words of Glass and Stanley, the assessment methods for endless and non-terminated cases give the same results on a daily basis. General sukupn_stu can name and renew the sukupn_st value of such signs. The reliability of the vibration to the general level є by the head office for the assessment of the characteristics of the general reliability for the characteristics of the vibration.

The main idea Mathematical statistics are based on reconsideration of those that, in addition to the introduction of all general sufficiency in large science establishments, is either practically unwise, or economically insufficient, some of the most vitally important materials. That in mathematical statistics stagnates vibrating pidhid, the principle of which is shown in the diagram in Fig. 1.2.

For example, for the technology of the formulation, the vibrations are randomized (simple and systematic), stratified, clustered (div. Rozdil 4).

Small. 1.2. Scheme of storing methods in mathematical statistics Згідно vibrating walk The test of mathematical and statistical methods can be carried out in the following way (div. Fig. 1.2):

o s general sukupnost, the powers that be methods to mold the vibrator- the type of ale is surrounded by a number of objects, up to which the previous methods are used;

o as a result of methods of precautions, experimental performances and tests of vibrating objects to recognize empirical data;

o processing empirical tributes for additional methods of descriptive statistics give indicators of vibration, which are called statisticians - as the name of the discipline, before speech;

o zasosovyuchi methods of statistical changes to statistician, find the parameters that characterize the power general sukupnost.

Application 1.1. Methods for assessing the stability of the level of knowledge (change X) a test of a randomized vibration was carried out by 3 students in volume n. Tests were revenged on m employees, who were judged by the system of points: "viconano" "- 1," not viconano "- 0.

3 randomized vibrating(Від English Random - vypadkovy) - tse representative vibіrka, as formed by the strategy of vypadkovy viprobuvan.

on івні past rockіv / h? Last solution:

o z'yasuvati the hypothesis of the type: "as long as the current results and testing will not be seen in advance, then it is possible to respect the knowledge of the students in the immutable, but initial process- stable ";

o formulate an adequate statistical hypothesis, for example, a null hypothesis H 0 about those that "the current average score X does not statistically look like the average indicator of past rockiv / h", so H 0: X = / g, against all kinds of alternative hypotheses X Ф ^;

o stay the empirical growth of the preliminaries X;

o visibility(If necessary) correlated calls, for example, between different X and the most important indicators, lines of regression;

o reconsider the validity of the empirical rise to the normal law;

o assessment of the value of point indicators and the most important interval of parameters, for example, average;

o the importance of criteria for converting statistical hypotheses;

o verification of statistical hypotheses on the basis of other criteria;

o formulate a decision about the statistical null hypothesis on the singing equal significance;

o go to the decision about acceptance or the conclusion of statistical null-hypotheses of interpretation of visnous as well as hypotheses of hypotheses;

o formulate changes in visnovka.

Also, as a result of the distribution of the procedure, the storage of statistical methods is based on three main blocks:

The transition from the object of reality to an abstract mathematical-statistical scheme, to induce the imovirnin model of manifestation, process, power;

Conducting rozrahunkovyh diy vasne mathematical methods within the framework of the iMovirn_n_model following the results of vim_ryuvan, thanksgiving, experiment and the formulation of statistical data;

Interpretation of statistical reports about the real situation and the acceptance of a general solution.

Statistical methods of processing and interpretation of data are directed to the theory of imperialities. The theory of imovings is the basis of the methods of mathematical statistics. Without the fundamental understanding of the laws of the theory of mathematical statistics, it is unfortunate that the visas of mathematical statistics are considered, and it means that they are primed for scientific and practical purposes.

So, to the managers of the descriptive statistics є the revision of the set of vibrating tributes to the system of indicators - statistics - frequency distribution, entering the central tendency and ministry, the efficiency of the connection is thin. However, statistics є characteristics, by day, specifically vibrating. Obviously, it is possible to develop vibrating rosters, vibrating mean, dispersion, etc., Ale podibny "analiz danih" can be combined with scientific and educational value. "Mechanic" transfer of any visnovka, broken on the basis of such indicators, on the other hand, is not correct.

In order for the possibility of transferring vibrational indicators, either for a little or for a larger extent, it is necessary to mathematically grounded. provision about the conformity and health of vibrating characteristics to the characteristics of the broader so-called general groups. Such provisions are based on theoretical approachesі diagrams connected with the world models of reality, for example, on an axiomatic basis, in the law of great numbers, etc. Only from the other side it is possible to transfer the power, which was established by the results of the analysis of the interconnected empirical information, either on the other or on the broadening of the scope. In such a rank, prompted, the laws of function, vicariousness of imovirnіsnyh models, є the subject of the mathematical area under the name "the theory of imovarnosti", a hundred sutta of statistical methods.

Thus, in mathematical statistics, there are two parallel rows of indicators: the first row, which can be used before practice (including vibrating indicators) and other, grounding on the theory (price indicators of an innovative model). For example, empirical frequencies, such as values ​​for vibrations, are suggested by an understanding of theoretical imaging; Vibrating average (practice) is shown mathematically (theory), etc. Moreover, in preliminaries the vibrating characteristics, as a rule, are the first ones. Smell to be rooted on the basis of precautions, vimіryuvan, experiments, in order to be able to pass a statistical assessment of the ability and efficiency, reconsideration of statistical hypotheses according to the goals of the success of the singers in the future.

Pitannya. Zavdannya.

1. Describe the main sections of mathematical statistics.

2. Who has the main idea of ​​mathematical statistics?

3. Describe the relationship of general and vibrating sukupnosti.

4. Explain the scheme of storing methods of mathematical statistics.

5. Introduce a transcript of the main works of mathematical statistics.

6. What are the main blocks of storage of statistical methods? Describe їх.

7. Open the links of mathematical statistics from the theory of imovarities.

Skin dosage in the field of vipadkovyh appearances to their roots and go to the experiment, in the past. Numeric tributes, which are taken when vivchenny some signs of a deyako ob'uktu, are called statistical... Statistical data є cob material of the past. In order for the stench to represent a science or practical value, it is necessary to process the methods of mathematical statistics.

Mathematical statistics- the price of scientific discipline, the subject of the development of methods of restoration, I will describe that analysis of statistical experimental data, taken as a result of the prevention of mass diseases.

Basic knowledge of mathematical statistics є:

viznachennya to the law of rozpodil vypadkovy values ​​or systems and vypadkovyh values;

reconsideration of the plausibility of hypotheses;

The value of undesirable parameters in the rosette.

All methods of mathematical statistics are based on the theory of definitions. However, by virtue of the specificity of the Viryshuvanih zavdan, the mathematical statistics are seen from the theory of imperialities in an independent area. If in the theory of imaging the given model of a manifestation is taken into account і it is carried out for the development of a possible real interruption of the manifestation (Fig. 1), then in the mathematical statistics the approaching theoretical model is taken into account (fig. 2).

Fig. 1. The fundamental problem of the theory of imovities

Fig. 2. The fundamental problem of mathematical statistics

The scientific discipline of mathematical statistics was developed simultaneously with the theory of imperialities. Mathematical apparatus of the science of motivation in the other half of the 19th century.

2. Generality and vibration.

For the introduction of statistical methods, an understanding of the general and vibrational sukupnosti is introduced. Have a zagalnyy vipadku pid general sukupnistyu the difference between the size of X and the function of the
... Vibrating sukupnistu chi vibrating volume for a given magnitude X is called typing
nezalezhnyh warnings of the tsієї magnitude, de bear the name of the vibrating value for the realization of the vipadkovo velichinX. In such a rank, it is possible to look at the number (as an experiment of conducting and vibrating) and as vyadkovі sizes(Before the experiment is carried out), the odd odor changes from vibrating to vibrating.

butt 1... For the purpose of determining the fallowness of the tree, 200 trees were taken from the top of the tree. In this vypadku obsyag vibirki n = 200.

Butt 2. As a result of cutting wood shavings on a circular saw blade, the 15th value of the pet robot was taken into account. In general, n = 15.

D
For this purpose, for the vibration data, it is necessary to judge about us the meaning of general sufficiency, about the vibration of the guilt correctly and imagine, so that the vibration is guilty but representative(Representative). The representativeness of the vibrator is easy to reach in the range of objects: the skin object of general sufficiency will be safe from the others.

Fig. 3. Demonstration of the representativeness of the vibra