Multinomial distribution as part of valuation of the number of failures
Authors: Novozhilov E. O. 
Published: March 2023

Abstract. Aim. The paper aims to examine the application of a multinomial distribution as part of valuation of the number of an object’s failures. It is assumed that the valuation is “based on past experience” (a statistical sample of the number of an object’s failures accumulated over several preceding evaluation periods).

Methods. The paper uses methods of system analysis, probability theory and mathematical statistics. The author analyses the primary indicators used to define the applied dependability indicators. It is noted that the valuation of such indicators based on statistical data for complex systems appears to be promising. The problem of valuation of the number of failures using a statistical sample is examined. The primary disadvantages of the used approaches are identified that are associated with errors in defining the average values of series, variation coefficients and asymmetry. It is shown that it is possible to solve the problem using the well-known combinatorics problem “on balls and boxes”, which leads to the use of a multinomial distribution. The paper examined the definition of probabilities for compositions and partitions of the number n into m parts, as well as the probabilities of a given number of balls being in a box with their maximum number. The author also considered formulas and algorithms that allow reducing the number of calculations in case of machine computation of the probabilities of a multinomial distribution. The feasibility of approximating a discrete distribution function by the Gumbel distribution is estimated. The paper demonstrates the feasibility of valuating the number of failures for a “segment” corresponding to a part (fraction) of an object’s dimension considered on a certain part (fraction) of the time interval. It also examines examples of valuating the number of failures for an object as a whole over a 1-month evaluation interval and for ½ of an object over a 12-month evaluation interval, while the total interval for which the statistical sample is accumulated is 72 months. The paper sets forth limitations on the application of the presented method and notes some of its possible advantages. In particular, it is noted that the statistical sample is only one implementation of the multinomial distribution, so it can be said that when applying the proposed method, the results of valuation are no longer affected by the presence of unlikely combinations of series values in the statistical sample. It is also noted that when applying the proposed method of valuating the number of failures, the obtained acceptable value will never be less than or equal to the average value of the statistical sample.

Results. Formulas have been obtained for calculating, based on partitions, of the discrete density number and the maximum distribution function of a multinomial distribution. The paper presents the results of algorithm analysis for machine computation. The results of applying some algorithms are presented. A formula is proposed for approximating the distribution function of the maximum of a multinomial distribution using the Gumbel distribution (for the largest values) using the method of moments. The author recommends a range of values of the estimated interval, in which the proposed method provides acceptable reliability of the results. The task of further research is defined.