Statistical Analysis of Normally Distributed Data with a Limited Scattering Interval of Values Converted by Direct g(x) = x2; cos x; ax and Inverse Functions
DOI:
https://doi.org/10.15407/ujpe67.5.346Keywords:
normal distribution, mathematical expectation, variance, random variables, calculation and transfer of errors, transformation of a random variable with elementary functionsAbstract
The work is devoted to the theoretical analysis of the correct application of the model of a continuous normally distributed random variable in the substantiation of the so-called error transfer formulas in the problem of a statistical processing of experimental data. Attention is paid to the role of limiting the scattering interval of values of a random variable subjected to nonlinear direct g(X) transformation by elementary functions X2; aX and cos X, as well as the inverse g-1(X) = √X, arccos X, loga X to them. The regularities of the statistical averaging of the data obtained by the disorder of the Taylor transform functions are studied. To confirm the validity of the obtained results, the method of quadratic functional optimization is used.
References
D.J. Hudson. Statistics. Lectures on Elementary Statistics and Probability (CERN, 1964).
Z. Pawlowsky. Wstep do Statystyki Matematycznej (Statystyk, 1967).
I.M. Kodolov, S.T. Khudyakov. Teoreticheskie osnovy verojtnostnyh metodov v inzhenerno-economicheskich zadachah. Funkcional'nye pereobrazovanij sluchajnyh velechyn i sluchajnye fynkcii (MADI, 1985).
V.I. Romanenko, N.V. Kornilovska. On the accuracy of error propagation calculations by analytic formulas obtained for the inverse transformation. Ukr. J. Phys. 64 (3), 217 (2019).
https://doi.org/10.15407/ujpe64.3.217
G.G. Rode. Propagation of the measurement errors andmeasured means of physical quantities for the elementaryfunctions x2 and √x Ukr. J. Phys. 62, 148 (2017).
https://doi.org/10.15407/ujpe62.02.0184
G.G. Rode. Propagation of measurement errors and measured means of a physical quantity for the elementary functions cos(x) and arccos(x) Ukr. J. Phys. 61, 345 (2016).
https://doi.org/10.15407/ujpe61.04.0345
G.G. Rode. Propagation of the measurement error and the measured mean of a physical quantity for elementary functions ax and loga x. J. Phys. 64 (5), 345 (2019).
https://doi.org/10.15407/ujpe64.5.371
P.S. Kosobutskyy. On the simulation of the mathematical expectation and variance of samples for gaussiandistributed random variables. Ukr. J. Phys. 62 (2), 827 (2017).
https://doi.org/10.15407/ujpe62.09.0827
P.S. Kosobutsky. Analytical relations for the mathematical expectation and variance of a standard distributed random variable subjected to the √X transformation. Ukr. J. Phys. 63, 215 (2018).
https://doi.org/10.15407/ujpe63.3.215
I.I. Gorban. Probability Theory and Mathematical Statistics for Scientists and Engineers (NAS of Ukraine, IPPMMiS, 2003).
E. Ng, M. Geller. A table o f integrals of the error functions. J. Research of the National Bureau of Standerds-B. Math. Sci. B 73 (1), 1 (1969).
https://doi.org/10.6028/jres.073B.001
From Web Resource: Table of Integrals. 2014 From http://integral-table.com
A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev. Integraly i Rjady. Elementarnye Funkcii (Nauka, 1981).
From Web Resource: http://mathworld.wolfram.com/GaussianIntegral.html
T. Lang. Twently errors even you can find in biomedical research articles. Croatian Medical J. 45 (4), 361 (2004).
Downloads
Published
How to Cite
Issue
Section
License
Copyright Agreement
License to Publish the Paper
Kyiv, Ukraine
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4. Duration.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5. Loyalty.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.
.png){kind=small}
