Appendix C: Cumulative Distribution Function of the Standard Normal Distribution

2005 ◽  
pp. 299-300
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Standard Normal

scholarly journals Deterministic Sampling from Univariate Normal Distributions with Sierpinski Space-Filling Curves

2021 ◽  
Author(s):  
Hime Oliveira
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Number ◽  
Probability Distributions ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Space Filling ◽  
Space Filling Curves ◽  
Deterministic Sampling ◽  
Standard Normal

This work addresses the problem of sampling from Gaussian probability distributions by means of uniform samples obtained deterministically and directly from space-filling curves (SFCs), a purely topological concept. To that end, the well-known inverse cumulative distribution function method is used, with the help of the probit function,which is the inverse of the cumulative distribution function of the standard normal distribution. Mainly due to the central limit theorem, the Gaussian distribution plays a fundamental role in probability theory and related areas, and that is why it has been chosen to be studied in the present paper. Numerical distributions (histograms) obtained with the proposed method, and in several levels of granularity, are compared to the theoretical normal PDF, along with other already established sampling methods, all using the cited probit function. Final results are validated with the Kullback-Leibler and two other divergence measures, and it will be possible to draw conclusions about the adequacy of the presented paradigm. As is amply known, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. That said, sequences resulting from this kind of procedure are not truly random. Even so, and to be coherent with the literature, the expression ”random number” will be used along the text to mean ”pseudo-random number”.

Asymptotic Approximation of an Integral Involving the Normal Distribution*

1986 ◽  
Vol 29 (2) ◽  
pp. 167-176 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Positive Constant ◽  
Asymptotic Approximation ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Normal Random Variable ◽  
Cumulative Distribution ◽  
Standard Normal Random Variable ◽  
Standard Normal

AbstractAn asymptotic approximation is obtained, as k → ∞, for the integralwhere Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.

scholarly journals A New One-term Approximation to the Standard Normal Distribution

2021 ◽  
pp. 381-385
Author(s):  
Ahmad Hanandeh ◽  
Omar Eidous
Keyword(s):  
Normal Distribution ◽  
Cumulative Distribution Function ◽  
Absolute Error ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Maximum Absolute Error ◽  
Practical Applications ◽  
Form Representation ◽  
Standard Normal ◽  
Term Approximation

This paper deals with a new, simple one-term approximation to the cumulative distribution function (c.d.f) of the standard normal distribution which does not have closed form representation. The accuracy of the proposed approximation measured using maximum absolute error (M.S.E) and the same criteria is used to compare this approximation with the existing one-term approximation approaches available in the literature. Our approximation has a maximum absolute error of about 0.0016 and this accuracy is sufficient for most practical applications.

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 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.

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],

Saddlepoint approximations for bivariate distributions

1990 ◽  
Vol 27 (03) ◽  
pp. 586-597 ◽  
Author(s):  
Suojin Wang
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Normal Distribution ◽  
Cumulative Distribution Function ◽  
Saddlepoint Approximation ◽  
Monte Carlo Study ◽  
Cumulative Distribution ◽  
Normal Distribution Function ◽  
Sample Mean ◽  
Saddlepoint Approximations

A saddlepoint approximation is derived for the cumulative distribution function of the sample mean of n independent bivariate random vectors. The derivations use Lugannani and Rice's saddlepoint formula and the standard bivariated normal distribution function. The separate versions of the approximation for the discrete cases are also given. A Monte Carlo study shows that the new approximation is very accurate.

scholarly journals Asymptotics for the time of ruin in the war of attrition

2017 ◽  
Vol 49 (2) ◽  
pp. 388-410 ◽  
Author(s):  
Philip A. Ernst ◽  
Ilie Grigorescu
Keyword(s):  
Distribution Function ◽  
Explicit Formula ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Fluid Limit ◽  
War Of Attrition ◽  
Time Of Ruin ◽  
Standard Normal ◽  
The Difference ◽  
The Time Of Ruin

AbstractWe consider two players, starting withmandnunits, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probabilityp(m,n) that the first player wins. Whenm~Nx0,n~Ny0, we prove the fluid limit asN→ ∞. Whenx0=y0,z→p(N,N+z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τNis established as (T- τN) ~N-βW1/β, β = ¼,T=x0+y0. Modulo a constant,W~ χ21(z02/T2).

scholarly journals Generation of pseudo-random numbers with the use of inverse chaotic transformation

2018 ◽  
Vol 16 (1) ◽  
pp. 16-22
Author(s):  
Marcin Lawnik
Keyword(s):  
Cumulative Distribution Function ◽  
Chaotic Maps ◽  
Continuous Distribution ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Random Numbers ◽  
Simulation And Modeling ◽  
Standard Normal ◽  
The Given

AbstractIn (Lawnik M., Generation of numbers with the distribution close to uniform with the use of chaotic maps, In: Obaidat M.S., Kacprzyk J., Ören T. (Ed.), International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH) (28-30 August 2014, Vienna, Austria), SCITEPRESS, 2014) Lawnik discussed a method of generating pseudo-random numbers from uniform distribution with the use of adequate chaotic transformation. The method enables the “flattening” of continuous distributions to uniform one. In this paper a inverse process to the above-mentioned method is presented, and, in consequence, a new manner of generating pseudo-random numbers from a given continuous distribution. The method utilizes the frequency of the occurrence of successive branches of chaotic transformation in the process of “flattening”. To generate the values from the given distribution one discrete and one continuous value of a random variable are required. The presented method does not directly involve the knowledge of the density function or the cumulative distribution function, which is, undoubtedly, a great advantage in comparison with other well-known methods. The described method was analysed on the example of the standard normal distribution.

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