Optimal Regularities of the Normal Distribution for Estimating the Sample Statistics of the Results of a Physical Experiment

  • P. Kosobutskyy National University “Lviv Polytechnic”


Basic probabilistic principles for the formation of the normal distribution for random fluctuations of physical quantities under the action of independent random factors on the physical system have been formulated. The emphasis is made on the integrated approach to the probabilistic statistical analysis of a sample of experimental results.

Keywords normal distribution, expectation, variance, random variables


1. G. Bohm, G. Zech. Introduction to Statistics and Data Analysis for Physicists (Deutsches Electronen-Synchrotron, 2010).
2. J.K. Patel, C.B. Read. Handbook of the Normal Distribution (Dekker, 1982) [ISBN: 0-8247-1541-1].
3. D.S. Sivia, J. Skilling. Data Analysis: A Bayesian Tutorial (Oxford Univ. Press, 2006) [ISBN: 978-0-19-856831-5].
4. M.V. Fock. Resolution of complex spectra into separate bands using theAlentsev method. Trudy Fiz. Inst. Akad. Nauk SSSR 63, 3 (1972) (in Russian).
5. A.S. Gusev. Probabilistic Methods in the Mechanics of Machines and Structures (Moscow State Technical University, 2009) (in Russian) [ISBN: 978-5-7038-3160-1].
6. E.Wichmann. Quantum Physics. Berkeley Physics Course (McGraw-Hill, 2010), Vol. 4 [ISBN: 0070048614].
7. F. Reif. Fundamentals Statistics and Thermal Physics (Waveland Press, 1965) [ISBN: 1-57766-612-7].
8. http://www.d.umn.edu/∼psiders/courses/chem5650/gaussviewtutorial /tutorial.html.
9. V.M. Zolotarev. One-Dimensional Stable Distributions (Amer. Math. Soc., 1986).
10. P.V. Korolenko, M.S. Maganova. Fundamentals of Statistical Methods in Optics (Universitetskaya Kniga, 2010) (in Russian) [ISBN: 978-5-91304-126-5].
11. F. Bardou, J.-Ph. Bouchaud, A. Aspect, C. Cohen-Tannoudji. L´evy Statistics and Laser Cooling. How Rare Events Bring Atoms to Rest (Cambridge Univ. Press, 2001).
12. O.N. Maslov. Stable Distributions and Their Application in Radio Engineering (Radio i Svyaz, 1994) (in Russian) [ISBN: 5-256-01187-1].
13. G. Samorodnitsky, M.S. Taqqu. Stable Non-Gaussian Random Processes (Chapman and Hall, 1994).
14. S. Holmes. Sums of Random Variables: Statistics 116 (Stanford University, 1998).
15. D. Titterington, A. Smith, U. Makov. Statistical Analysis of Finite Mixture Distributions (Wiley, 1985) [ISBN: 0-471-90763-4].
16. N.A. Carreira-Perpi˜n´an. Mode-finding for mixture of Gaussian distributions. IEEE Trans. Pattern Analys. Mach. Intell. 22, 1318 (2000).
17. Ya.A. Fomin, G.R. Tarlovskii. Statistical Theory of. Pattern Recognition (Radio i Svyaz, 1986) (in Russian).
18. J.A. Hartigan. Distribution Problems in Clustering. In Classification and Clustering, edited by J. Van Ryzin (Academic Press, (1977), p. 45.
19. N.N. Aprausheva, I.A. Gorlach, A.A. Zhelnin, S.V. Sorokin. An experiment on automated statistical recognition of clouds. J. Comput. Math. Math. Phys. 38, 1715 (1998).
20. M. Podrygalo, O. Isakova, A. Korobko. A new way to evaluate the coincidence of the results of theoretical and experimental studies Metrolog. Prylady 5, 48 (2017) (in Ukrainian).
21. D.Z. Shmatko, O.V. Kochneva. Calculation of the probability of defect determination in pieces by non-destructive control. Matematych. Model. No. 1, 51 (2017) (in Ukrainian).
22. P. Kosobutskyy. On the simulation of the mathematical expectation and variance of samples for Gaussian-distributed random variables. Ukr. Fiz. Zh. 61, 827 (2017) (in Ukrainian).
23. J.R. Stedinger, R.M. Vogel, E. Foufoula-Georgiou. Frequency anlysis of extreme events. In Handbook of Applied Hydrology, Part 3, edited by D. Maidment (McGraw-Hill, 1993), ch. 18 [ISBN: 978-0-07-171177-7].
24. A. Hald. Maximum likelihood estimation of the parameters of a normal distribution which is truncated at a known point. Scand. Actuar. J. No. 1, 119 (1949).
25. B. Gnedenko, I. Ushakov. Probabilistic Reliability Engineering (Wiley Interscience, 1995) [ISBN: 0-471-30502-2].
26. V.R. Matvievsky. Reliability of Engineering Systems (MGIEM, 2002) (in Russian) [ISBN: 5-230-22198-4].
27. A. O’Connor, M. Modarres, A. Mosleh. Probability Distributions Used in Reliability Engineering (CRE University of Maryland, 2016) [ISBN: 978-0-9966468].
28. J. Xinzhang, L. Tao. An empirical formula for yield estimation from singly truncated performance data of qualified
semiconductor devices. J. Semicond. 33, 125008 (2012).
29. K. Gu, X. Jia, H. You, T. Liang. The yield estimation of semiconductor products based on truncated samples. Int. J. Metrol. Qual. Eng. 4, 215 (2013).
30. A. Einstein. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Ann. der Phys. 17, 549 (1905).
31. A. Einstein, M. Smoluchowski. Brownian Motion. Collection of Papers (ONTI-GROTL, 1936) (in Russian).
32. H. Risken, T. Frank. The Fokker–Planck Equation: Methods of Solutions and Applications (Springer, 1989) [ISBN: 0-387-50498-2].
33. I.I. Gikhman, A.V. Skorokhod. The Theory of Stochastic Processes I (Springer, 2004) [ISBN: 3-540-20284-6].
34. E. Ng, M. Geller. A table of integrals of the error functions. J. Res. Natl. Bureau Stand. B 73B, 1 (1969).
35. Sh. Sulaberidze. Methods for Analyzing and Processing Measured Quantity Values (Baltian State Technical University, 2013) (in Russian) [ISBN: 978-5-85546-742-0].
36. E. Inman, E. Bradley. The owerlapping coefficient as a measure of the agreement between probability distributions and point estimation of the overlap of two normal densities. Commun. Stat. Theor. Methods 18, 3851 (1989).
37. Z. Karian, E. Dudewicz. Handbook of Fitting Statistical Distributions with R (Taylor and Francis, 2011) [ISBN: 978-1-58488-711-9].
38. D.C. Dowson, B.V. Landau. The Fr´echet distance between multivariate normal distributions. J. Multivar. Anal. 12, 450 (1982).
39. M. Varanasi, B. Aazhang. Parametric generalized Gaussian density estimation. J. Acoust. Soc. Am. 86, 1404 (1989).
40. S. Nadarajah. A generalized normal distribution. J. Appl. Stat. 32, 685 (2005).
41. J. Behboodian. On a mixture of normal distributions. Biometrica 57, 215 (1970).
42. N.N. Aprausheva, S.V. Sorokin. Notes on Gaussian Mixtures (Computation Center of the Russian Academy of Sciences, 2015) (in Russian).
43. B. Voigtlander. Scanning Probe Microscopy. Atomic Force Microscopy and Scanning Tunneling Microscopy (Springer, 2015) [ISBN: 978-3-662-45239-4].
44. N.M. Sergeev. NMR Spectroscopy (Moscow State University, 1981) (in Russian).
45. E. Rutherford, F. Soddy. A comparative study of the radioactivity of radium and thorium. Phil. Mag. 5, 445 (1903).
46. T. Akhromeeva, S. Kurdyumov, G. Malinetskii, A. Samarskii. Structures and Chaos in Nonlinear Media (Fizmatlit, 2007) (in Russian) [ISBN: 978-5-9221-0887-4].
47. M. Schroder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Freeman, 1991) [ISBN: 0-7167-2136-8].
48. A.A. Nikitin. Statistical Methods for Distinguishing Geophysical Anomalies (Nedra, 1979) (in Russian).
How to Cite
Kosobutskyy, P. (2018). Optimal Regularities of the Normal Distribution for Estimating the Sample Statistics of the Results of a Physical Experiment. Ukrainian Journal Of Physics, 63(7), 645. doi:10.15407/ujpe63.7.645
General physics