Istrazivanja i projektovanja za privreduJournal of Applied Engineering Science


DOI: 10.5937/jaes0-35413 
This is an open access article distributed under the CC BY 4.0
Creative Commons License

Volume 20 article 964 pages: 582-589

Mohammad M. Hamasha*
Department of Industrial Engineering, Faculty f Engineering, The Hashemite University, P.O.Box 330127, Zarqa 13133, Jordan Business Department

Abdulaziz Ahmed
Department of Health Services Administration, School of Health Professions, The University of Alabama at Birmingham, Birmingham, Alabama, USA

Haneen Ali
Health Services Administration Program, Auburn University, Auburn, AL, USA; Department of Industrial Engineering, Auburn University, Auburn, AL, USA

Sa'd Hamasha
Department of Industrial Engineering, Auburn University, Auburn, AL, USA

Faisal Aqlan
Industrial Engineering Department, University of Louisville, Louisville, KY, USA

The Gaussian or normal distribution is vital in most areas of industrial engineering, including simulation. For example, the inverse of the Gaussian cumulative density function is used in all simulation software (e.g., ARENA, ProModel) to generate a group of random numbers that fit Gaussian distribution. It is also used to estimate the life expectancy of new devices. However, the Gaussian distribution that is truncated from the left side is not defined in any simulation software. Estimation of the expected life of used devices needs left-sided truncated Gaussian distribution. Additionally, very few works examine generating random numbers from left-sided truncated Gaussian distribution. A high accuracy mathematical-based approximation to the left-sided truncated Gaussian cumulative density function is proposed in the current work. Our approximation is built based on Polya’s approximation of the Gaussian cumulative density function. The current model is beneficial to approximate the inverse of the left-sided truncated Gaussian cumulative density function to generate random variates, which is necessary for simulation applications.

View article

1. Tunno, B. J., Michanowicz, D. R., Shmool, J. L., Kinnee, E., Cambal, L., Tripathy, S., Gillooly, S., Roper, C., Chubb, L., Clougherty, J. E. (2016). Spatial variation in inversion-focused vs 24-h integrated samples of PM2. 5 and black carbon across Pittsburgh, PA. Journal of exposure science & environmental epidemiology, vol. 26, no. 4, 365-376.

2. Fischer, M., & Jakob, K. (2016). pTAS distributions with application to risk management. Journal of Statistical Distributions and Applications, vol. 3, no. 1, 1-18.

3. Cha, J., Cho, B. R. (2015). Classical statistical inference extended to truncated populations for continuous process improvement: test statistics, P‐values, and confidence intervals. Quality and Reliability Engineering International, vol. 31, no. 8, 1807-1824.

4. Ross, S. R. (2014). Introduction to probability models, 11th Ed. Elsevier, Oxford, UK

5. Zhang, X., Shen, C., Cheng, P., Li, Q. (2017). An image-processing based method for the measurement of the film thickness of a swirl atomizer. Journal of Visualization, vol. 20, no. 1, 1-5.

6. Harrison, D., Sutton, D., Carvalho, P., Hobson, M. (2015). Validation of Bayesian posterior distributions using a multidimensional Kolmogorov–Smirnov test. Monthly Notices of the Royal Astronomical Society, vol. 451, no. 3, 2610-2624.

7. Mahajan, A., Tatikonda, S. (2015). An algorithmic approach to identify irrelevant information in sequential teams. Automatica, vol. 61, 178-191.

8. Davidson, R. (2015). Computing, the bootstrap and economics. Canadian Journal of Economics/Revue canadienne d'économique, vol. 48, no. 4, 1195-1214.

9. Zuverink, A., (2015) Surface roughness scattering of electrons in bulk mosfets. Thesis Submitted to the University of Wisconsin-Madison, Madision, USA

10. Fang, X., Li, J., Wong, W. K., Fu, B. (2016). Detecting the violation of variance homogeneity in mixed models. Statistical methods in medical research, vol. 25, no. 6, 2506 2520.

11. Dutta, S., Misra, I. S. (2014). Error Analysis of 2-tierM-ary Star QAM Modulation in Shadowed Fading Channels. International Journal of Computer Applications, vol. 88, no. 1, 9-16.

12. Büyükkaracığan, N. (2014). Determining the best fitting distributions for minimum flows of streams in Gediz Basin. International Journal of Civil and Environmental Engineering, vol. 8, no. 6, 417-422.

13. Moheghi, H., Niaki, S. T. A., Bootaki, B., Bakhshesh, D. (2017). On the effect of inducted negative correlation rate for beta acceptance–rejection algorithms. Communications in Statistics-Simulation and Computation, vol. 46, no. 3, 2152-2167.

14. Hörmann, W., Leydold, J. (2014). Generating generalized inverse Gaussian random variates. Statistics and Computing, vol. 24, no. 4, 547-557.

15. Lijoi, A., Prünster, I. (2014). Discussion of “On simulation and properties of the stable law” by L. Devroye and L. James. Statistical methods & applications, vol. 23, no. 3, 371-377.

16. Zhu, H., Dick, J. (2014). Discrepancy bounds for deterministic acceptance-rejection samplers. Electronic Journal of Statistics, vol. 8, no. 1, 678-707.

17. Xi, B., Tan, K. M., Liu, C. (2013). Logarithmic transformation-based gamma random number generators. Journal of Statistical Software, vol. 55, 1-17.

18. Hung, Y. C., Chen, W. C. (2017). Simulation of some multivariate distributions related to the Dirichlet distribution with application to Monte Carlo simulations. Communications in Statistics-Simulation and Computation, vol. 46, no. 6, 4281-4296.

19. Romano, P. K. (2015). An algorithm for generating random variates from the Madland–Nix fission energy spectrum. Computer Physics Communications, vol. 187, 152-155.

20. Favaro, S., Nipoti, B., Teh, Y. W. (2015). Random variate generation for Laguerre-type exponentially tilted $\alpha $-stable distributions. Electronic Journal of Statistics, 9, no. 1, 1230-1242.

21. Bowling, S. R., Khasawneh, M. T., Kaewkuekool, S., Cho, B. R. (2009). A logistic approximation to the cumulative normal distribution. Journal of Industrial Engineering and Management, vol. 2, no. 1, 114-127.

22. Jawitz, J. W. (2004). Moments of truncated continuous univariate distributions. Advances in water resources, vol. 27, no. 3, 269-281.

23. Khasawneh, M. T., Bowling, S. R., Kaewkuekool, S., Cho, B. R. (2004). Tables of a truncated standard normal distribution: A singly truncated case. Quality Engineering, vol. 17, no. 1, 33-50.

24. Khasawneh, M. T., Bowling, S. R., Kaewkuekool, S., Cho, B. R. (2005). Tables of a truncated standard normal distribution: A doubly truncated case. Quality Engineering, vol. 17, no. 2, 227-241.

25. Kim, T. M., Takayama, T. (2003). Computational improvement for expected sliding distance of a caisson-type breakwater by introduction of a doubly-truncated normal distribution. Coastal Engineering Journal, vol. 45, no. 3, 387-419.

26. Makarov, Y. V., Loutan, C., Ma, J., De Mello, P. (2009). Operational impacts of wind generation on California power systems. IEEE transactions on power systems, vol. 24, no. 2, 1039-1050.

27. Cha, J., Cho, B. R., Sharp, J. L. (2013). Rethinking the truncated normal distribution. International Journal of Experimental Design and Process Optimisation, vol. 3, no. 4, 327-363.

28. Hamasha, M., Al, H., Hamasha, S., Ahmed, A. (2021). A mathematical approximation to left-sided truncated normal distribution based on Hart's model. Journal of Applied Engineering Science, vol. 19, no. 4, 1049-1055.

29. Burkardt, J. (2014). The truncated normal distribution. Department of Scientific Computing Website, Florida State University, 1, 35.