Skip to main content
Log in

The evaluation of bivariate normal probabilities for failure of parallel systems

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Albers W, Kallenberg WCM (1994) A simple approximation to the bivariate normal distribution with large correlation coefficient. J Multivar Anal 49:87–96

    Article  MathSciNet  MATH  Google Scholar 

  • Antia HM (2012) Numerical methods for scientists and engineers, 3rd edn. Hindustan Book Agency, New Delhi, pp 177–187

    Book  MATH  Google Scholar 

  • Cox DR, Wermuth N (1991) A simple approximation for bivariate and trivariate normal integrals. Int Stat Rev 59(2):263–269

    Article  MATH  Google Scholar 

  • Daley DJ (1974) Computation of bi-and tri-variate normal integrals. J R Stat Soc Ser C 23:435–438

    Google Scholar 

  • Ditlevsen O (1979) Narrow reliability bounds for structural system. J Struct Mech 7(4):453–472

    Article  Google Scholar 

  • Divgi DR (1979) Calculation of univariate and bivariate normal probability functions. Ann Stat 7:903–919

    Article  MathSciNet  MATH  Google Scholar 

  • Dong YG, Lu HT, Guo B (2013) A method for improving computational accuracy of reliability based on the theory of the minimum cut set. Chin J Mech Eng 49(20):184–191 ((in Chinese))

    Article  Google Scholar 

  • Dong YG, Zhang HM, He LG, Wang C, Wang MH (2019) The computation of standard normal distribution integral in any required precision based on reliability method. Commun Stat Theory Methods 48(6):1517–1528

    Article  MathSciNet  MATH  Google Scholar 

  • Drezner Z (1978) Computation of the bivariate normal integral. Math Comput 32:277–279

    Article  MathSciNet  MATH  Google Scholar 

  • Drezner Z, Wesolowsky GO (1990) On the computation of the bivariate normal integral. J Stat Comput Simul 35:101–107

    Article  MathSciNet  Google Scholar 

  • Fayed HA, Atiya AF (2014) A novel series expansion for the multivariate normal probability integral based on Fourier series. Math Comput 83:2385–2402

    Article  MathSciNet  MATH  Google Scholar 

  • Genz A (2004) Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Stat Comput 14:251–260

    Article  MathSciNet  Google Scholar 

  • Gong JX, Zhao GF (1996) An approximate algorithm for bivariate normal integral. Chin J Comput Mech 13(4):494–4999 ((in Chinese))

    MathSciNet  Google Scholar 

  • Harkness WL, Godambe AV (1976) Inequalities for tail probabilities for the multivariate normal distribution. Commun Stat Theory Methods A5(7):689–692

    Article  MathSciNet  MATH  Google Scholar 

  • Hong HP (1999) An approximation to bivariate and trivariate normal integrals. Civ Eng Environ Syst 16:115–127

    Article  Google Scholar 

  • Hutchinson TP, Lai CD (1990) Continuous bivariate distributions: emphasizing applications. Rumsby, Adelaide

    MATH  Google Scholar 

  • Kim J (2013) The computation of bivariate normal and t probabilities, with application to comparisons of three normal means. Comput Stat Data Anal 58:177–186

    Article  MathSciNet  MATH  Google Scholar 

  • Lin JT (1995) A simple approximation for the bivariate normal integral. Probab Eng Inf Sci 9:317–321

    Article  MathSciNet  MATH  Google Scholar 

  • Lozy MDME (1982) Simple computation of a bivariate normal integral arising from a problem of misclassification with applications to the diagnosis of hypertension. Commun Stat Theory Methods 11(19):2195–2205

    Article  Google Scholar 

  • Mee RW, Owen DB (1983) A simple approximation for bivariate normal probabilities. J Qual Technol 15(2):72–75

    Article  Google Scholar 

  • Milton RC (1972) Computer evaluation of the multivariate normal integral. Technometrics 14(4):881–889

    Article  MATH  Google Scholar 

  • Monhor D (2011) A new probabilistic approach to the path criticality in stochastic PERT. Cent Eur J Oper Res 19:615–633

    Article  MathSciNet  MATH  Google Scholar 

  • Monhor D (2013) Inequalities for correlated bivariate normal distribution function. Probab Eng Inf Sci 27:115–123

    Article  MathSciNet  MATH  Google Scholar 

  • Moskowitz H, Tsai HT (1989) An error-bounded polynomial approximation for bivariate normal probabilities. Commun Stat Simul Comput 18(4):1421–1437

    Article  MATH  Google Scholar 

  • Owen DB (1956) Tables for computing bivariate normal probabilities. Ann Math Stat 27:1075–1090

    Article  MathSciNet  MATH  Google Scholar 

  • Pandey MD (1998) An effective approximation to evaluate multinormal integrals. Struct Saf 20:51–67

    Article  Google Scholar 

  • Rodgers JL, Nicewander WA (1988) Thirteen ways to look at the correlation coefficient. Am Stat 42(1):59–66

    Article  Google Scholar 

  • Sathe YS, Lingras SR (1980) A note on the inequalities for tail probability of the multivariate normal distribution. Commun Stat Theory Methods A9(7):711–715

    Article  MathSciNet  MATH  Google Scholar 

  • Sándor Z, András P (2004) Alternative sampling methods for estimating multivariate normal probabilities. J Econ 120:207–234

    Article  MathSciNet  MATH  Google Scholar 

  • Terza JV, Welland U (1991) A comparison of bivariate normal algorithms. J Stat Comput Simul 39:115–127

    Article  Google Scholar 

  • Verma AK, Ajit S, Karanki DR (2016) Reliability and safety engineering, 2nd edn. Springer, Heidelberg, pp 257–292

    Google Scholar 

  • Vijverberg WPM (1997) Monte Carlo evaluation of multivariate normal probabilities. J Econ 76:281–307

    Article  MathSciNet  MATH  Google Scholar 

  • Willink R (2004) Bounds on the bivariate normal distribution function. Commun Stat Theory Methods 33(10):2281–2297

    Article  MathSciNet  MATH  Google Scholar 

  • Young JC, Minder CE (1974) Algorithm AS 76: an integral useful in calculating non-central t and bivariate normal probabilities. J R Stat Soc Ser C 23:455–457

    Google Scholar 

  • Yuan XX, Pandey MD (2006) Analysis of approximations for multinormal integration in system reliability computation. Struct Saf 28:361–377

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Yuge Dong or Qingtong Xie.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, Y., Xie, Q., Ding, S. et al. The evaluation of bivariate normal probabilities for failure of parallel systems. Stat Papers 63, 1585–1614 (2022). https://doi.org/10.1007/s00362-021-01282-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-021-01282-9

Keywords

Navigation