The evaluation of bivariate normal probabilities for failure of parallel systems

Yuge Dong (), Qingtong Xie (), Shuguang Ding, Liangguo He and Hu Wang
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Yuge Dong: Hefei University of Technology
Qingtong Xie: Hefei University of Technology
Shuguang Ding: Hefei University of Technology
Liangguo He: Hefei University of Technology
Hu Wang: Hefei University of Technology

Statistical Papers, 2022, vol. 63, issue 5, No 10, 1585-1614

Abstract: Abstract A method for computing the joint failure probability of a parallel system consisting of two linear limit state equations is given to indirectly obtain the value of the integral of the standard bivariate normal distribution by using the conclusion that the joint failure probability of the parallel system is equal to the value of the double integral. In the two-dimensional standard normal coordinate system, some circles whose centres are all at the coordinate origin are used to divide the two-dimensional standard sample space into a number of pairwise disjoint sub-sample spaces and obtain a number of pairwise disjoint sub-failure domains. According to the probability theory, the probability of any sub-failure domain can be expressed by using a sub-sample space probability and a conditional probability. Based on the total probability formula, the joint failure probability can be obtained by the sum of the probabilities of the sub-failure domains because the sub-sample spaces or the sub-failure domains are pairwise disjoint. By introducing a random variable obeying the Rayleigh distribution, it is possible to compute probabilities of the sub-sample spaces accurately. The formulae of computing the conditional probability are derived. The main parameters related to the computation of the joint failure probability, such as the minimum and maximum radii, and the number of the dividing circles, are discussed to make the computation process easy and the computed result meet a required precision. Examples show that it is possible and significant for the method in the paper to complete the computation of the standard bivariate normal distribution integral with high accuracy.

Keywords: Reliability computation; Linear limit state equation; Standard normal coordinate system; Joint failure domain; Standard bivariate normal distribution integral; Sub-sample space; Sub-failure domain; Conditional probability (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00362-021-01282-9

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