One-term Approximation for Normal Distribution Function

Omar Eidous,Samar Al-Salman

doi:10.13189/ms.2016.040102

Abstract

This paper presents a one-term approximation to the cumulative normal distribution functions. The absolute maximum error of the proposed approximation is 0.0018 less than 0.003 of Polya's approximation. Comparisons between the proposed approximation and the different approximations with one-term that stated in the literature are given.

Highlights

Normal distribution is considered as one of the most important distribution functions in statistics because it is simple to handle analytically
It is well known that the cumulative normal distribution function does not have closed form representation
There is no closed-form solution available for the above integral and its value is usually found from the tables of the cumulative normal distribution for different values of xx

Summary

Introduction

Normal distribution is considered as one of the most important distribution functions in statistics because it is simple to handle analytically. A number of approximate functions for a cumulative normal distribution function have been reported in the research community (See for example, Aludaat and Alodat [1]; Johnson et al [2]; Bailey [3] and Polya [4]). This paper proposed a simple approximation with one-term for Φ(xx) , xx ≥ 0 The idea of this proposed approximation is similar to that of Polya (1945) and Aludaat and Alodat [1]. The absolute maximum error of the proposed approximation is less than that of Aludaat and Alodat [1] and Polya [4]. The one-term approximation of Aludaat and Alodat [1] has absolute maximum error 0.002 less than 0.003 of Polya's one-term approximation. Johnson et al [2] and Yerukala and Boiroju [7])

The Proposed Approximation

Comparison