Estimating the Number of Terms in the Sum of Independent Random Variables when Modeling Gaussian Random Variables
DOI:
https://doi.org/10.52575/2687-0932-2022-49-3-546-557Keywords:
sum of random variables, average sample, number of terms, characteristic function, Maclaurin series, accuracy, error, integralAbstract
One of the most important problems in probability theory and statistics is the estimation of the number of terms of the central limit theorem necessary for the sum to have a normal probability distribution law. This problem becomes especially relevant for any previously unknown distribution laws of random variables. To solve the problem, the apparatus of characteristic functions is used. The characteristic function is represented by a complex Maclaurin series. The representation of the residual term of the series in the form of Cauchy is used. An estimate of the error of such a representation is obtained. An analytical expression is derived for the distribution density of the average sample of observations. An algorithm has been developed to determine the required number of observations depending on the accuracy of the assessment. To explain the operation of the algorithm, an example of the practical implementation of the developed method is shown. The obtained simulation results for estimating the required number of terms are summarized in a table. For clarity, the results are presented in the graph. The statements proved in the work can be used in multivariate data analysis, systems for diagnostics, monitoring, and statistical control of manufactured products.
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