Some Features of the Direct and Inverse Transformation of Random Variables

: pp. 50 - 61
Lviv Polytechnic National University, Ukraine
Lviv Polytechnic National University

Always actual tasks of obtaining and processing experimental results in complex systems. Random obstacles (errors), measurement errors, imperfections and limitations of mathematical models and data processing algorithms can change the appearance of the distribution and lead to incorrect use of algorithms, for example, as is the case with Kalman filtering in control systems. Complex methods for the identification of distribution laws require the study of quantum systems, natural phenomena, environmental, biological, etc. processes, which are characterized by the presence of singularities and  multimodality of distributions. Therefore, it is often  not recommended to apply  separate distribution laws to simulate probabilistic experimental data distributions, but a generalized distribution as a single statistical system, which known distributions include as individual partial cases. Thus, the generalized gamma distribution includes Rayleigh, Maxwell, Weibull, Levy, Hi-Square distributions, which are widely used in applied problems associated with statistical methods of physical processes research, remote sensing, in the theory of reliability, for describing the dispersion composition of particles fragmentation and calculation of the efficiency of phase separation in gas-liquid streams.

  1. Stace E. A generalization of the gamma distribution. Ann.Math.Statistiics.1962, 33, P. 1187–1192.
  2. Королев В. Ю., Крылов В. А., Кузьмин В. Ю. Устойчивость конечных смесей обобщенных гамма- распределений относительно возмущений параметров. Информатика и ее применения. 2011, Т. 5, вып.1, С. 31–38.
  3. Коузов П. А. Основы анализа дисперсионныого состава промышленных пылей и измельченных материалов. Л.: Химия, 1987, 264 с.
  4. Subbotin M. T. On the law of frequency of error // Математический сборник, 1923. Т. 31. Вып. 2. С. 296–301.
  5. Новицкий П. В., Зограф И. А. Оценка погрешностей результатов измерений. Л.: Энергоатомиздат, 1991. 
  6. Гонсалес Р. Цифровая обработка изображений / Р. Гонсалес, Р. Вудс. М.: Техносфера, 2005. 1072 с.
  7. Goodman, J. W. Speckle Phenomena in Optics: Theory and Applications / J. W. Goodman. Roberts & Company, Publishers, Englewood, CO, 2006. 387 p.
  8. Teran-Bobadilla E., MendezE. A study of the fluctuations of the optical properties of a turbid media through Monte Carlo method. arXiv:1507.01522v1 [physics.optics] 6 July, 2015.
  9. Кравцов Ю. А., Рытов С. М., Татарский В. И. Статистические проблемы в теории дифракции. Успехи физических наук. Т. 115, No. 2, 1975, с. 239–262.
  10. Honerkamp J. Statistical Physics. An Advanced Approach and Applocations. Web-enhanced with Problems and Solutions.Springer-Verlag Berlin Heidelberg, 2002.
  11. Suhir E. Applied Probability for Engineers and Scientistics (McGraw-Hill Companies, 1997.
  12. Papoulis A.Probability, Random Variables, and Stochastic Processes.1991, McGraw-Hill, 1991.
  13. Матвиевский В. Р. Надежность технических систем. М.: Московский государственный интситут электроники и математики, 2002. 113 с. 
  14. Kimber A. C., Jeynes C. An Application of the Truncated Two-Piece Normal Distribution to the Measurement of Depths of Arsenic Implants in Silicon. Journal of the Royal Statistical Society. Series C (Applied Statistics) Vol. 36, No. 3 (1987), pp. 352–357.
  15. Gu K., Jia X., You H., Liang T. The yield estimation of semiconductor products based on truncated samples. Int. J. Metrol. Qual. Eng. 4, рр. 215–220 (2013).
  16. Xinzhang J., Tao L. An empirical formula for yield estimation from singly truncated performance data of qualified semiconductor devices. Journal of Semiconductors. Vol. 33, No. 12, 2012.

  17. Holický M. Functions of Random Variables. In: Introduction to Probability and Statistics for Engineers. Springer, Berlin, Heidelberg (2013) 18.
  18. Роде
  19. Kosobutsky P. Analytical relations for the mathematical expectation and variance of a standardly distributed random variable subjected to transformation. Ukr. J. Phys. vol. 63(3), P. 215–219, 2018.
  20. Mande J. The Statistical Analysis of Experimental Data (New York: Dover Publications,Inc,1964) [ISBN0-486-64666-1]. 
  21. Koski T. Lecture Notes. Probabiity and Random Processes at KTN for sf2940 Probability Theory (Stockholm: KTN Royal Institute of Technology,2017)

  22. Л. де Бройль. Соотношения неопределенностей Гейзенберга и вероятностная интерпретация волновой механіки. М.: Мир, 1986.

  23. NIST Handbook of Mathematical Functions. Ed. Olver F., Lozier D., Boisvert R., Clark C. NIST National Institute of Standart and Technology U.S. Department of Commerce and Cambridge University Press, 2010, p. 163.

  24. Bohm G., Zech G. Іntroduction to Statistics and data Analysis for Physics. Verlag Deutsches Elektronen- Synchrotron.

    Hald A. Statistical Theory with Engineering Applications. New York-London, 1952;

  25. Hald A. Maximum Likelihood Estimation of the Parameters of a Normal Distribution which is Truncated at a Known Point. Scandinavian Actuarial Journal . Vol. 1949, 1949 – Issue 1.