A MATHEMATICAL APPROXIMATION TO LEFT-SIDED TRUNCATED NORMAL DISTRIBUTION BASED ON HART’S MODEL

Left-sided truncated distributions (LSTD) have been found in different situations in the industry. For example, the life distribution of used devices is left-sided truncated distribution. Moreover, if a lower specification exists without the upper specification limit, the product distribution is truncated from the left side. Left-sided truncated normal distributions (LSTND) is a special case where the original distribution is normal. LSTND characteristics, as well as cumulative densities and probabilities can be difficult to employ manually, with most practitioners relying largely on specialized (and expensive) software. In many cases, practitioners are against purchasing software, as they are often limited in the number of estimations. The paper will provide an accurate and straightforward approximation to the cumulative density of LSTND. Hart’s normal distribution is simplified and used as a foundation of this model. The maximum absolute error for the curve at different truncation points (i.e., ZL) over the definition range (i.e., [zL: ∞]) is as follows: 0.004303 for ZL=-4, 0.00432 for ZL=-3, 0.00449 for ZL=-2, 0.005727 for ZL=-1, and 0.0106 for ZL=0. Even the maximum errors are very ignorable in probability applications. Further, it is rare to find a truncation point of higher than -2 in the industry.

Download Full-text

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

Engineering, Technology & Applied Science Research ◽

10.48084/etasr.988 ◽

2017 ◽

Vol 7 (1) ◽

pp. 1382-1386

Author(s):

M. M. Hamasha

Keyword(s):

Normal Distribution ◽

Distribution Density ◽

Standard Normal Distribution ◽

Maximum Deviation ◽

Density Functions ◽

Truncated Normal Distribution ◽

Life Distribution ◽

Standard Normal ◽

Mathematical Approximation ◽

Truncated Normal

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.

Download Full-text

Standard tables of truncated standard normal distribution using a new summarizing method

World Journal of Engineering ◽

10.1108/wje-02-2017-0041 ◽

2018 ◽

Vol 15 (2) ◽

pp. 216-247

Author(s):

Mohammad M. Hamasha ◽

Mohammad Al-Rabayah ◽

Faisal Aqlan

Keyword(s):

Normal Distribution ◽

Linear Interpolation ◽

Standard Normal Distribution ◽

Truncated Normal Distribution ◽

Content Type ◽

Normal Distributions ◽

Population Distributions ◽

Wide Range ◽

Standard Normal ◽

Truncated Normal

Purpose The single- and double-sided truncated normal distributions have been used in a wide range of engineering fields. However, most of the previous research works have focused primarily on the non-truncated population distributions. The authors present reference tables to estimate the values of density and cumulative density functions of truncated normal distribution for practitioners. Finally, the authors explain how to use the tables to estimate other properties, such as mean, median and variance. The purpose of this paper is to provide an efficient method to summarize tables, and furthermore, to provide readers with statistical tables on truncated standard normal distribution. Design/methodology/approach A new methodology is developed to summarize the tables with ordered values. The introduced method allows for the reduction of the number of pages required for such tables into a reasonable level by using linear interpolation. Moreover, it allows for the estimation of the required truncation values accurately with an error value less than 0.005. Findings The data in the tables can be summarized into a significantly reduced amount. The new summarized table can be designed for any number of pages and/or level of error wanted. However, with reducing the level of error, the number of pages increases and vice versa. Originality/value The value of this work is through two major points. First, all provided summarized tables in the literature are for single-sided and symmetry truncation cases. However, there is no attempt to summarize the tables of the asymmetry truncation normal distribution due to the requirement of huge number of pages. In this paper, the case of asymmetry truncation is included. Second, the methodology provided in this research can be used to summarize similar large tables.

Download Full-text

Features of accumulation of amounts of measurement error

Geodesy and Cartography ◽

10.22389/0016-7126-2017-928-10-58-63 ◽

2017 ◽

Vol 928 (10) ◽

pp. 58-63 ◽

Cited By ~ 1

Author(s):

V.I. Salnikov

Keyword(s):

Measurement Error ◽

Confidence Interval ◽

Normal Distribution ◽

Confidence Level ◽

Measurement Errors ◽

Truncated Normal Distribution ◽

Truncated Normal ◽

The Law ◽

Normal Law

The initial subject for study are consistent sums of the measurement errors. It is assumed that the latter are subject to the normal law, but with the limitation on the value of the marginal error Δpred = 2m. It is known that each amount ni corresponding to a confidence interval, which provides the value of the sum, is equal to zero. The paradox is that the probability of such an event is zero; therefore, it is impossible to determine the value ni of where the sum becomes zero. The article proposes to consider the event consisting in the fact that some amount of error will change value within 2m limits with a confidence level of 0,954. Within the group all the sums have a limit error. These tolerances are proposed to use for the discrepancies in geodesy instead of 2m*SQL(ni). The concept of “the law of the truncated normal distribution with Δpred = 2m” is suggested to be introduced.

Download Full-text