The optimal estimates of the probability of a random event

One of the most important problems of designing production systems is to ensure their reliability. At the same time, it is meant not only the technical reliability of technological equipment, but also the influence of external and internal random factors that lead to failures of the production process.

In most works related to the study of regularities of random processes in production and other technical systems, the known axiomatics is used. The corresponding analytical tool allows to solve the problem of assessing the probability of production tasks (for example, daily schedule of steel smelting or monthly production plan).

The known formula for estimating the probability of some random event, which determines the specified probability as the ratio of the number of successful experiments to the number of all experiments, is usually accepted as an axiom.

There is no doubt that this axiom is fair, as it is always confirmed by experience. However, it is interesting to obtain this confirmation analytically.

This paper presents an analytical conclusion of this formula.

According to the frequency axiomatics of probability theory, the probability of some event p* is determined by the ratio of the number of realizations of this event y in a series of n independent tests to the number of these tests.

The value of y has a binomial distribution, which at sufficiently large n by Laplace’s theorem tends to normal with the same parameters. Almost normal law can be used already at n > 15 .

We write the evaluation model p* as the relative frequency, that is, the ratio y to n.

The distribution law of p* as a linear function of y is also asymptotically normal.

The numerical characteristics of the distribution of p* are calculated by known formulas for linear functions of random variables.

Questions arise: is the estimate of p* optimal and how to estimate its variance to then compute a*[p*]? To answer these questions, an optimization problem is formulated, for which the Lagrange multiplier method is used.

The solution of this problem has shown that the axiomatic values of the 1/n multipliers used are optimal in the sense of the minimum variance of the random event probability estimate.

1. Anan’ev B.I. Optimizing estimation of a statistically undefined system. Avtomatika i telemekhanika = Automation and remote. 2018. No. 1. Pp. 18-32. (In Russ.)

2. Efrosinin D.V., Farkhadov M.P., Stepanova N.V. Study of a Controllable Queueing System with Unreliable Heterogeneous Servers. Avtomatika i telemekhanika = Automation and remote. 2018. No. 2. Pp. 80-105. (In Russ.)

3. Naumov V.A., Samuilov K.E. Analysis of Networks of the Resource Queuing Systems. Avtomatika i telemekhanika = Automation and remote. 2018. No. 5. Pp. 59-68. (In Russ.)

4. Kang YS., Sobol V.R. Asymptotic confidence interval for conditional probability at decision making. Avtomatika i telemekhanika = Automation and remote. 2017. No. 10. Pp. 130-138. (In Russ.)

5. Gamkrelidze N.G. On One Property of Symmetrized Distributions. Theory Probab. Appl. 2018. Vol. 62. No. 1. Pp. 55-57. DOI: 10.1137S0040585X97T988484

6. Bavrin I.I. Teoriya veroyatnostei i matematicheskaya statistika [Probability theory and mathematical statistics]. Moscow: Vysshaya shkola, 2005. 160 p. (In Russ.)

7. Venttsel’ E.S., Ovcharov L.A. Zadachi i uprazhneniya po teorii veroyatnostei [Problems and exercises in probability theory]. Moscow: Izdatel’skii tsentr «Akademiya», 2003. 448 p. (In Russ.)

8. Vilenkin N.Ya., Vilenkin A.N., Vilenkin P.A. Kombinatorika [Combinatorics]. Moscow: FIMA, MTsNMO, 2006. 400 p. (In Russ.)

9. Vukolov E.A., Efimov A.V., Zemskov V.N., Pospelov A.S. Sbornik zadach po matematike dlya vtuzov, ch. 4 [Collection of problems in mathematics for technical colleges]. Moscow: Fizmatlit, 2004. 432 p. (In Russ.)

10. Gmurman V. E. Rukovodstvo k resheniyu zadach po teorii veroyatnostei i matematicheskoi statistike [Guide to solving problems in probability theory and mathematical statistics]. Moscow: Vysshaya shkola, 2004. 404 p. (In Russ.)

11. Gmurman V. E. Teoriya veroyatnostei i matematicheskaya statistika [Probability theory and mathematical statistics] Moscow: Vysshaya shkola, 2009. 478 p. (In Russ.)

12. Gnedenko B.V. Kurs teorii veroyatnostej: uchebnik dlya studentov matematicheskix specialnostej universitetov [Course of probability theory: textbook for students of mathematical specialties of universities]. Moscow: Izdatel'stvo LKI, 2007. 447 p. (In Russ.)

13. Kibzun A.I. Teoriya veroyatnostei imatematicheskaya statistika. Bazovyi kurs s primerami i zadachami [Probability theory and mathematical statistics. Basic course with examples and tasks]. Moscow: Fizmatlit, 2002. 224 p. (In Russ.)

14. Kibzun A.I., Goryainova E.R., Naumov A.V. Teoriya veroyatnostei i matematicheskaya statistika. Bazovyi kurs s primerami i zadachami [Probability theory and mathematical statistics:a basic course with examples and problems]. Moscow: Fizmatlit, 2007. 231 p. (In Russ.)

15. Kremer N.Sh. Teoriya veroyatnostei i matematicheskaya statistika [Probability theory and mathematical statistics]. Moscow: UNITY-DANA, 2004. 573 p. (In Russ.)

16. Maksimov Yu.D. Veroyatnostnye razdely matematiki [Probabilistic areas of mathematics]. St. Petersburg: Ivan Fedorov, 2001.592 p. (In Russ.)

17. Goryainov V.B., Pavlov I.V., Tsvetkova G.M., Teskin O.I. Matematicheskaya statistika [Mathematical statistics]. Moscow: Iedatel’stvo MGTU im. N.E. Baumana, 2001.424 p. (In Russ.)

18. Pis’mennyi D.T. Konspekt lektsii po teorii veroyatnostei, matematicheskoi statistike i sluchainym protsessam [Lecture Notes on probability theory, mathematical statistics and random processes]. Moscow: Airis-press, 2008. 288 p. (In Russ.)

19. Pugachev VS. Teoriya veroyatnostei i matematicheskaya statistika [The theory of probability and mathematical statistics]: Moscow: Fizmatlit, 2002. 496 p. (In Russ.)

20. Sidnyaev N.I. Teoriya planirovaniya eksperimenta i analiz statisticheskikh dannykh [Theory of experiment planning and analysis of statistical data]. Moscow: Yurait, 2012. 399 p. (In Russ.)

21. Pechinkin A.V., Teskin O.I., Tsvetkova G.M. and etc. Teoriya veroyatnostei [Probability theory]. Moscow: Iedatel’stvo MGTU im. N.E.Baumana, 2004. 456 p. (In Russ.)

22. Informatika[Informatics]. Ed. prof. N.V Makarovoi. St. Petersburg: Piter, 2010. 768 p. (In Russ.)

23. Vtyurin V.A. Komp’yuternye tekhnologii v oblasti avtomatizatsii i upravleniya [Computer technologies in the field of automation and control]. St. Petersburg: SPbGLTU, 2011. 103 p. (In Russ.)

24. Norenkov I.P. Osnovy avtomatizirovannogo proektirovaniya [Fundamentals of computer-aided design]. Moscow: Iedatel’stvo MGTU im. N.E.Baumana, 2002. 333 p. (In Russ.)

25. Gnedenko B.V. Kurs teorii veroyatnostei: uchebnik dlya studentov matematicheskikh spetsial’nostei universitetov [Course of probability theory: textbook for students of mathematical specialties of universities]. Moscow: Izdatel’stvo LKI, 2007. 447 p.