A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the Cadwell Approximation Model


  • M. M. Hamasha Department of Engineering Management, Prince Sultan University, Saudi Arabia
Volume: 7 | Issue: 1 | Pages: 1382-1386 | February 2017 | https://doi.org/10.48084/etasr.988


In the case that life distribution of new devices follows the normal distribution, the life distribution of the same brand used devices follows left-sided truncated normal distribution. In spite of many mathematical models being available to approximate the normal distribution density functions, there is a few work available on modeling/approximating the density functions of left-sided truncated normal distribution. This article introduces a high accuracy mathematical model to approximate the cumulative density function of left-sided truncated standard normal distribution defined on the range of [truncation point (ZL): ∞]. The introduced model is derived from the Cadwell approximation of the normal cumulative density. The accuracy level change with Z score is discussed in details. The maximum deviation of the model results, from the real results for the whole region of [-∞<Z<-2:∞], is 0.006877.


truncated normal distribution, normal distribution, mathematical approximation


Download data is not yet available.


M. M. Hamasha, “Practitioner advice: approximation of the cumulative density of left-sided truncated normal distribution using logistic function and its implementation in Microsoft Excel”, Qual. Eng., 2016 DOI: https://doi.org/10.1080/08982112.2016.1196373

S. R. Bowling, M. T. Khasawneh, S. Kaewkuekool, B. R. Cho, “A logistic approximation to the cumulative normal distribution”, J. Ind. Eng. Manage., Vol. 2, No. 1, pp. 114-127, 2009 DOI: https://doi.org/10.3926/jiem.2009.v2n1.p114-127

J. H. Cadwell, “The bivariate normal integral”, Biometrika, Vol. 38, pp. 475-479, 1951 DOI: https://doi.org/10.1093/biomet/38.3-4.475

H. Vazquez-Leal, R. Castaneda-Sheissa, U. Filobello-Nino, A. Sarmiento-Reyes, J. S. Orea, “High accurate simple approximation of normal distribution integral”, Math. Prob. Eng., Vol. 2012, Article ID 124029, pp. 1-22, 2016 DOI: https://doi.org/10.1155/2012/124029

A. Choudhury, “A simple approximation to the area under standard normal curve”, Math. Stat., Vol. 2, No. 3, pp. 147-149, 2014

A. J. Luis, “An Approximation to the Probability Normal Distribution and Its Inverse”, Ingeniería, Investigación y Tecnología, Vol. 16, No. 4, pp. 605-611, 2015 DOI: https://doi.org/10.1016/j.riit.2015.09.012

X. Ke, J. Hou, T. Chen, “Notice of retraction research on action reliability for ammunition swing device of large caliber gun based on doubly truncated normal distribution”, International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering (QR2MSE), July 15-18, 2013 [retracted] DOI: https://doi.org/10.1109/QR2MSE.2013.6625630

J. Pender, The truncated normal distribution: Applications to queues with impatient customers”, Oper. Res. Lett., Vol. 43 No. 1, pp. 40-45, 2015 DOI: https://doi.org/10.1016/j.orl.2014.10.008

X. D. Sun, X. X. Tao, C. Q. Liu, “Finite fault modelling for the Wenchuan earthquake using hybrid slip model with truncated normal distributed source parameters”, Appl. Mech. Mater., Vol. 256-259, pp. 2161-2167, 2013 DOI: https://doi.org/10.4028/www.scientific.net/AMM.256-259.2161

H. Beucher, D. Renard, “Truncated Gaussian and derived methods”, In Press, C. R. Geosci., Vol. 348, No. 7, pp. 510–519, 2016 DOI: https://doi.org/10.1016/j.crte.2015.10.004

R. Mukerjee, S. H. Ong, “Variance and covariance inequalities for truncated joint normal distribution via monotone likelihood ratio and log-concavity”, J. Multi. Var. Anal., Vol. 139, pp. 1-6, 2015 DOI: https://doi.org/10.1016/j.jmva.2015.02.010

W. C. Horrace, “On ranking and selection from independent truncated normal distributions”, J. Econometrics, Vol. 126, pp. 335-354, 2005 DOI: https://doi.org/10.1016/j.jeconom.2004.05.005

J. J. Sharples, J. C. V. Pezzey, Expectations of linear functions with respect to truncated multinormal distributions–With applications for uncertainty analysis in environmental modelling”, Environ. Modell. Software, Vol. 22, No. 7, pp. 915-923, 2007 DOI: https://doi.org/10.1016/j.envsoft.2006.05.021

C. E. Ebeling, An introduction to reliability and maintainability, 2nd Ed. Waveland Pr Inc, USA. 2009


How to Cite

M. M. Hamasha, “A Mathematical Approximation of the Left-sided Truncated Normal Distribution Using the Cadwell Approximation Model”, Eng. Technol. Appl. Sci. Res., vol. 7, no. 1, pp. 1382–1386, Feb. 2017.


Abstract Views: 632
PDF Downloads: 532

Metrics Information
Bookmark and Share