Simulation of statistical mean and variance of normally distributed random values, transformed by nonlinear functions $\sqrt{|X|}$ and $\sqrt{X}$

This paper presents theoretical studies of formation regularities for the statistical mean and variance of normally distributed random values with the unlimited argument values subjected to nonlinear transformations of functions $\sqrt{|X|}$ and  $\sqrt{X}$.  It is shown that for nonlinear square root transformation of a normally distributed random variable, the integrals of higher order mean $n>1$ satisfy the inequality $\overline{(y-\overline{Y})^n}\neq 0$.  On the basis of the theoretical research, the correct boundaries $m,\sigma \to \infty$ of error transfer formulas are suggested.

  1. Weisstein E. W.  Cauchy Distribution.  From MathWorld-A Wolfram Web Resource.
  2. Hudson D.  Lectures on probability theory and elementary statistics.  Geneva, CERN (1963).
  3. Suhir E.  Applied Probability for Engineers and Scientistics.  McGraw-Hill Companies (1997).
  4. Papoulis A.  Probability, Random Variables, and Stochastic Processes.  McGraw-Hill (1991).
  5. Kodolov I. M., Khudyakov S. T.  Teoreticheskie osnovy verojtnostnyh metodov v inzhenerno-economicheskich zadachah.  Funkcional'nye pereobrazovanij sluchajnyh velechyn i sluchajnye fynkcii.  Moskva, MADI (1985), (in Russian).
  6. Romanenko V. I., Kornilovska N. V.  On the accuracy of error propagation calculations by analytic formulas obtained for the inverse transformation.  Ukrainian Journal of Physics. 64 (3), 217222 (2019).
  7. Leone F. C., Nelson L. S., Nottingham R. B.  The folded normal distribution.  Technometrics. 3 (4), 543550 (1961).
  8. Gui W., Chen P.-H., Wu H.  A Folded Normal Slash Distribution and its Applications to Non-negative Measurements.  Journal of Data Science. 11 (2), 231247 (2013).
  9. Rode G. G.  Propagation of the Measurement Errors and Measured Means of Physical Quantities for The Elementary Functions $x^2$ and $\sqrt{x}$.  Ukrainian Journal of Physics. 62 (2), 184191 (2017).
  10. Fotiadis D., Scheuring S., Müller S. A., Engel A., Müller D. J.  Imaging and manipulation of biological structures with the AFM.  Micron. 33 (4), 385397 (2002).
  11. Vattulainen I., Ala-Nissila T., Kanakaala K.  Physical tests for random numbers simulations.  Physical Review Letters. 73 (19), 25132516 (1994).
  12. Lang T.  Twently Statistical Errors Even You Can Find in Biomedical Research Articles.  Croatian Medical Journal. 45 (4), 361–370 (2004).
  13. Prudnikov A. P., Brychkov Yu. A., Marichev O. I.  Integraly i rjady. Elementarnye funkcii.  Moskva, Nauka (1981), (in Russian).
  14. Ng E. W., Geller M.  A Table o f Integrals of the Error Functions.  Journal of Research of the National Bureau of Standerds – B. Mathematical Sciences. 73B (1), 1–20 (1969).
  15. From Web Resource: Table of Integrals.  2014 From http://integral-table.com.
  16. Kosobutskyy P. S.  On the simulation of the mathematical expectation and variance of samples for gaussian-distributed random variables.  Ukrainian Journal of Physics. 62 (2), 827–831 (2017).
  17. Kosobutskyy P. S.  Analytical relations for the mathematical expectation and variance of a standard distributed random variable subjected to the $\sqrt{x}$ transformation.  Ukrainian Journal of Physics. 63 (3), 215–219 (2018).