IDEAS home Printed from https://ideas.repec.org/a/spr/stpapr/v63y2022i5d10.1007_s00362-021-01282-9.html
   My bibliography  Save this article

The evaluation of bivariate normal probabilities for failure of parallel systems

Author

Listed:
  • 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)

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.

Suggested Citation

  • Yuge Dong & Qingtong Xie & Shuguang Ding & Liangguo He & Hu Wang, 2022. "The evaluation of bivariate normal probabilities for failure of parallel systems," Statistical Papers, Springer, vol. 63(5), pages 1585-1614, October.
    • Handle: RePEc:spr:stpapr:v:63:y:2022:i:5:d:10.1007_s00362-021-01282-9
    DOI: 10.1007/s00362-021-01282-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00362-021-01282-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00362-021-01282-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. J. C. Young & Ch. E. Minder, 1974. "An Integral Useful in Calculating Non‐Central T and Bivariate Normal Probabilities," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 23(3), pages 455-457, November.
    2. Albers, W. & Kallenberg, W. C. M., 1994. "A Simple Approximation to the Bivariate Normal Distribution with Large Correlation Coefficient," Journal of Multivariate Analysis, Elsevier, vol. 49(1), pages 87-96, April.
    3. Yuge Dong & Haimeng Zhang & Liangguo He & Can Wang & Minghui Wang, 2019. "The computation of standard normal distribution integral in any required precision based on reliability method," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(6), pages 1517-1528, March.
    4. Kim, Jongphil, 2013. "The computation of bivariate normal and t probabilities, with application to comparisons of three normal means," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 177-186.
    5. D. J. Daley, 1974. "Computation of Bi‐ and Tri‐Variate Normal Integrals," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 23(3), pages 435-438, November.
    6. Vijverberg, Wim P. M., 1997. "Monte Carlo evaluation of multivariate normal probabilities," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 281-307.
    7. Ajit Kumar Verma & Srividya Ajit & Durga Rao Karanki, 2016. "Reliability and Safety Engineering," Springer Series in Reliability Engineering, Springer, edition 2, number 978-1-4471-6269-8, January.
    8. Davaadorjin Monhor, 2011. "A new probabilistic approach to the path criticality in stochastic PERT," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 19(4), pages 615-633, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Erik Plug & Wim Vijverberg, 2003. "Schooling, Family Background, and Adoption: Is It Nature or Is It Nurture?," Journal of Political Economy, University of Chicago Press, vol. 111(3), pages 611-641, June.
    2. Krzysztof S. Targiel & Maciej Nowak & Tadeusz Trzaskalik, 2018. "Scheduling non-critical activities using multicriteria approach," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(3), pages 585-598, September.
    3. Haoying Wang & Guohui Wu, 2022. "Modeling discrete choices with large fine-scale spatial data: opportunities and challenges," Journal of Geographical Systems, Springer, vol. 24(3), pages 325-351, July.
    4. Ziegler, Andreas, 2002. "Simulated Classical Tests in the Multiperiod Multinomial Probit Model," ZEW Discussion Papers 02-38, ZEW - Leibniz Centre for European Economic Research.
    5. Z. I. Botev, 2017. "The normal law under linear restrictions: simulation and estimation via minimax tilting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 125-148, January.
    6. Adelchi Azzalini & Monica Chiogna, 2004. "Some results on the stress–strength model for skew-normal variates," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 315-326.
    7. Ziegler, Andreas, 2001. "Simulated z-tests in multinomial probit models," ZEW Discussion Papers 01-53, ZEW - Leibniz Centre for European Economic Research.
    8. Zsolt Sandor, 2009. "Multinomial discrete choice models (in Russian)," Quantile, Quantile, issue 7, pages 9-19, September.
    9. Rennings, Klaus & Ziegler, Andreas & Zwick, Thomas, 2001. "Employment changes in environmentally innovative firms," ZEW Discussion Papers 01-46, ZEW - Leibniz Centre for European Economic Research.
    10. Penttinen, Jussi-Pekka & Niemi, Arto & Gutleber, Johannes & Koskinen, Kari T. & Coatanéa, Eric & Laitinen, Jouko, 2019. "An open modelling approach for availability and reliability of systems," Reliability Engineering and System Safety, Elsevier, vol. 183(C), pages 387-399.
    11. Bui, Ha & Sakurahara, Tatsuya & Pence, Justin & Reihani, Seyed & Kee, Ernie & Mohaghegh, Zahra, 2019. "An algorithm for enhancing spatiotemporal resolution of probabilistic risk assessment to address emergent safety concerns in nuclear power plants," Reliability Engineering and System Safety, Elsevier, vol. 185(C), pages 405-428.
    12. Paulsen, Jostein & Lunde, Astrid & Skaug, Hans Julius, 2008. "Fitting mixed-effects models when data are left truncated," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 121-133, August.
    13. Sandor, Zsolt & Andras, P.Peter, 2004. "Alternative sampling methods for estimating multivariate normal probabilities," Journal of Econometrics, Elsevier, vol. 120(2), pages 207-234, June.
    14. Amaral, Andrea & Abreu, Margarida & Mendes, Victor, 2014. "The spatial Probit model—An application to the study of banking crises at the end of the 1990’s," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 415(C), pages 251-260.
    15. Arias, Carlos & Cox, Thomas L., 1999. "Maximum Simulated Likelihood: A Brief Introduction For Practitioners," Staff Papers 12662, University of Wisconsin-Madison, Department of Agricultural and Applied Economics.
    16. Sándor, Z. & András, P., 2003. "Alternate Samplingmethods for Estimating Multivariate Normal Probabilities," Econometric Institute Research Papers EI 2003-05, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    17. Andreas Ziegler, 2010. "Individual Characteristics and Stated Preferences for Alternative Energy Sources and Propulsion Technologies in Vehicles: A Discrete Choice Analysis," CER-ETH Economics working paper series 10/125, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich.
    18. Carlos Arias & THOMAS L. COX, 1999. "Maximum Simulated Likelihood: A Brief Introduction for Practitioners," Wisconsin-Madison Agricultural and Applied Economics Staff Papers 421, Wisconsin-Madison Agricultural and Applied Economics Department.
    19. Liesenfeld, Roman & Richard, Jean-François, 2010. "Efficient estimation of probit models with correlated errors," Journal of Econometrics, Elsevier, vol. 156(2), pages 367-376, June.
    20. Heijnen, P. & Samarina, A.. & Jacobs, J.P.A.M. & Elhorst, J.P., 2013. "State transfers at different moments in time," Research Report 13006-EEF, University of Groningen, Research Institute SOM (Systems, Organisations and Management).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stpapr:v:63:y:2022:i:5:d:10.1007_s00362-021-01282-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.