scholarly journals High Accurate Simple Approximation of Normal Distribution Integral

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Roberto Castaneda-Sheissa ◽  
Uriel Filobello-Nino ◽  
Arturo Sarmiento-Reyes ◽  
Jesus Sanchez Orea
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Relative Error ◽  
Gaussian Distribution ◽  
Error Function ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Normal Distribution Function ◽  
Standard Normal ◽  
Gaussian Integral

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 . 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.

ON THE GENERALISATION OF SIDEL’NIKOV’S THEOREM TO -ARY LINEAR CODES

2019 ◽  
Vol 101 (1) ◽  
pp. 157-162
Author(s):  
YILUN WEI ◽  
BO WU ◽  
QIJIN WANG
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Linear Codes ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Binary Codes ◽  
Normal Distribution Function ◽  
Code Length ◽  
Standard Normal Distribution Function ◽  
Standard Normal

We generalise Sidel’nikov’s theorem from binary codes to $q$-ary codes for $q>2$. Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by $\unicode[STIX]{x1D6F7}(z)$ the standard normal distribution function, we show that $|A(z)-\unicode[STIX]{x1D6F7}(z)|$ is bounded above by a term which tends to $0$ when the code length tends to infinity.

scholarly journals Calculation of the throwing angle of a fertilizer centrifugal device as functions of random coordinates of feeding point

2020 ◽  
Vol 175 ◽  
pp. 05016 ◽  
Author(s):  
Vasiliy Chernovolov ◽  
Lyudmila Kravchenko ◽  
Alla Nikitina ◽  
Vladimir Litvinov
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Angular Speed ◽  
Standard Normal Distribution ◽  
Normal Distribution Function ◽  
Angle Distribution ◽  
Particle Movement ◽  
The Matrix ◽  
Standard Normal ◽  
Centrifugal Device

Liquid fertilizers fed into centrifugal device are spread along angle and radius of feed under action of blades. This article describes how to calculate throwing angular characteristics using Mathcad. The package consists of four programs. Program Mf is intended for calculation of probability density of supply point coordinates under assumption of bivariant normal distribution of system r, γ, which are specified in the form of vectors. The result of the calculation is displayed as matrix Mf. The program Mα calculates the throwing angle for all combinations r, γ.. To calculate the throwing angle, the method of solving differential equations of particle movement along the blade of the device with input data was used: Radius of the disk R, angular speed ω, coefficient of friction of fertilizers on the blade f. The program Ms extracts from the matrix Mf elements Corresponding to a throw angle less than a given number A. The program F (A) sums the elements of the matrix Ms. We obtained the values of the throw angle distribution function by multiplying the resulting sum by the intervals of vectors r and γ. The calculated throwing angle distribution function is approximated by the standard normal distribution function.

A Characterization of Deviation from Normality Under Certain Moment Assumptions

1966 ◽  
Vol 9 (4) ◽  
pp. 509-514
Author(s):  
W.R. McGillivray ◽  
C.L. Kaller
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Standard Normal Distribution ◽  
Standard Normal ◽  
Image Position

If Fn is the distribution function of a distribution n with moments up to order n equal to those of the standard normal distribution, then from Kendall and Stuart [1, p.87],

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