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Pearson distributions of sum of single distributed independent random variables.
Golik Felix Valentinovich

Doctor of Technical Science

Professor, Novgorod Branch of the Russian Academy of National Economy and Public Administration under the auspices of the President of the Russian Federation

173003, Russia, Novgorodskaya oblast', g. Velikii Novgorod, ul. Germana, 31, aud. 401


The article is devoted to working out the constructive method of approximation the sum of independent random variables with the same distribution by Pearson curves. The summation theory was and still is one of the key parts of the theory of probability. The limiting theorems are proven within this theory, and they allow one to understand which frequencies may be used for the approximation for the sum so random values with large m. At the same time the approximation error is evaluated by the admissible error. However, in most practical cases the number of the summed values is not large, so the admissible error evaluation may not be sufficiently precise. The purpose of the study is to develop a constructive method for the approximation of the frequency function for the spread of the final sum of  the independent random values with the same frequency. The Pearson curves are then used as approximative frequencies. Such an approximation lacks the defects related to the application of limiting theorems. It is applicable for any number of summed accidental frequencies m>1. The calculated ratios for the initial moments of the final sum of independent random variables are obtained. It is shown that the parameters of the Pearson curves for the sum m of random variables are related by simple ratios with the corresponding parameters of the summed value. The solution used in order to achieve the goal is based upon the moments method. Thе author offers a recursion formula for calculating the starting moments of for the sum of independent random values, allowing to find the central moments of the sum, as well as the parameters for the Pearson curves.  It is proven that there's a dependency between the distance from the point of  The exact expression for the distance from  the point, corresponding to the distribution of the sum of the random variables in the coordinate system of Pearson parameters to the point (0, 3), corresponding to the normal distribution is found. By the distance value, one can indirectly assess the possibility of applying normal approximation. The author studies the possibility for the approximation of Pearson curves with normal distribution. An approximate formula for estimating the error in approximating the sum  of random variables by normal distribution is given. The author provides examples of approximations for the distribution of the sum of random variables are found, which are often met in statistical radio engineering tasks. The reference materials include complete formulae for the key types of Pearson curves. All the obtained results are applicable for any random variables having finite first four initial moments. The correctness of the conclusions is confirmed by numerical calculations performed in the MathCad program.

Keywords: probability measure, recursive algorithm, sum of random variables, normal distribution, moments of a random variable, density function, Pearson curve, random variable, approximation error, method of the moments



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