Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2012, Article ID 124029, 22 pages
Research Article

High Accurate Simple Approximation of Normal Distribution Integral

1Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Cto. Gonzalo Aguirre Beltrán S/N, Zona Universitaria Xalapa, 91000 Veracruz, VER, Mexico
2Electronics Department, National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No.1, 72840 Tonantzintla, PUE, Mexico

Received 8 September 2011; Revised 15 October 2011; Accepted 18 October 2011

Academic Editor: Ben T. Nohara

Copyright © 2012 Hector Vazquez-Leal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The integral of the standard normal distribution function is an integral without solution and represents the probability that an aleatory variable normally distributed has values between zero and 𝑥 . The normal distribution integral is used in several areas of science. Thus, this work provides an approximate solution to the Gaussian distribution integral by using the homotopy perturbation method (HPM). After solving the Gaussian integral by HPM, the result served as base to solve other integrals like error function and the cumulative distribution function. The error function is compared against other reported approximations showing advantages like less relative error or less mathematical complexity. Besides, some integrals related to the normal (Gaussian) distribution integral were solved showing a relative error quite small. Also, the utility for the proposed approximations is verified applying them to a couple of heat flow examples. Last, a brief discussion is presented about the way an electronic circuit could be created to implement the approximate error function.