2016, Vol. 3, No. 3. - go to content...

Метаданные этой статьи так же доступны на русском языке DOI: 10.15862/05TS316 (https://doi.org/10.15862/05TS316)

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Stolyarov V.V., Shchegoleva N.V. [On the limits to applicability of the normal distribution law instead of the binomial distribution while the statistical processing of discrete integer values] Russian Journal of Transport Engineering, 2016, Vol. 3, No. 3. Available at: https://t-s.today/PDF/05TS316.pdf (in Russian). DOI: 10.15862/05TS316

## On the limits to applicability of the normal distribution law instead of the binomial distribution while the statistical processing of discrete integer values

Stolyarov Viktor Vasilevich
Yuri Gagarin state technical university of Saratov, Russia, Saratov
E-mail: stolyarov_v_v@mail.ru

Shchegoleva Natalia Vyacheslavovna
Yuri Gagarin state technical university of Saratov, Russia, Saratov
E-mail: Shegoleva123@mail.ru

Abstract. In the transport construction, as in all fields of human activity, there are problems associated with the statistical processing of discrete integer values varying sequentially by one. Such problems very often arise when constructing the densities of distribution of the discrete random variables to assess the risk of adverse event occurrence in the «driver — car — road — environment» system. It is known that the probability of the particularly discrete and integer value occurrence is described by the binomial distribution, the use of which for a large number of tests becomes very time-taking and lengthy process, associated with a large input of given data, varying, as already mentioned, successively by one, into the specialized computer programs. Therefore it is very important when handling the applied problems to be able to use the method of transition from the binomial distribution to the normal law derived by A. Moivre as early as in 1733, and suitable for solution of the assigned problems. However, without studies described in this article this is difficult to do, since the existing method has no formulae for determining the lower and upper limits of integration, which are used when determining the probability of hit of a random variable to a dedicated site. Besides derivation of these formulae there were developed the mathematical expressions to determine the serial number of the maximum variable, wherein the normal and binomial distributions give the same probability. These mathematical expressions take into account the value of mode deviation from the mathematical expectation in the binomial distribution law. Finally, using a solution of the modern theory of probability, the author shows a boundary condition for the applicability of the normal distribution instead of the binomial one with an absolute error of probability equal to 0.05 of the probability obtained by the binomial law. The article presents the main conclusions.

Keywords: probability density; Laplace function; function of normal distribution; binomial distribution of discrete integer values; mathematical expectation and mean value; mean-square deviation; independent variables or factors; critical parameters; limit to applicability of the normal law instead of the binomial distribution 