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”.

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.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

scholarly journals Transmuted Probability Distributions: A Review

2020 ◽  
pp. 83-94
Author(s):  
Md. Mahabubur Rahman ◽  
Bander Al-Zahrani ◽  
Saman Hanif Shahbaz ◽  
Muhammad Qaiser Shahbaz
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Quantile Function ◽  
Cumulative Distribution ◽  
Functional Composition ◽  
Inverse Cumulative Distribution Function ◽  
Family Of Distributions

Transmutation is the functional composition of the cumulative distribution function (cdf) of one distribution with the inverse cumulative distribution function (quantile function) of another. Shaw and Buckley(2007), first apply this concept and introduced quadratic transmuted family of distributions. In this article, we have presented a review about the transmuted families of distributions. We have also listed the transmuted distributions, available in the literature along with some concluding remarks.

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.

A Pedagogical Note on Asymptotic Normality of a Two-Sample Approximate Pivot for Comparing Means

2021 ◽  
pp. 000806832097948
Author(s):  
Nitis Mukhopadhyay
Keyword(s):  
Pointwise Convergence ◽  
Cumulative Distribution Function ◽  
Asymptotic Theory ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Sample Sizes ◽  
Standard Normal ◽  
Practical Standpoint ◽  
Thinking Process

A two-sample pivot for comparing the means from independent populations is well known. For large sample sizes, the distribution of the pivot is routinely approximated by a standard normal distribution. The question is about the thinking process that may guide one to rationalize invoking the asymptotic theory. In this pedagogical piece, we put forward soft statistical arguments to make users feel more at ease by suitably indexing the sample sizes from a practical standpoint that would allow a valid interpretation and understanding of pointwise convergence of the pivot's cumulative distribution function (c.d.f.) to the c.d.f. of a standard normal random variable.

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 Alpha-Skew Generalized t Distribution

2015 ◽  
Vol 38 (2) ◽  
pp. 371-384 ◽  
Author(s):  
Sukru Acitas ◽  
Birdal Senoglu ◽  
Olcay Arslan
Keyword(s):  
Cumulative Distribution Function ◽  
Real Life ◽  
Likelihood Estimation ◽  
Point Of View ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Density Functions ◽  
Life Problems ◽  
Standard Normal ◽  
T Distribution

<p>The alpha-skew normal (ASN) distribution has been proposed recently in the literature by using standard normal distribution and a skewing approach. Although ASN distribution is able to model both skew and bimodal data, it is shortcoming when data has thinner or thicker tails than normal. Therefore, we propose an alpha-skew generalized t (ASGT) by using the generalized t (GT) distribution and a new skewing procedure. From this point of view, ASGT can be seen as an alternative skew version of GT distribution. However, ASGT differs from the previous skew versions of GT distribution since it is able to model bimodal data sest as well as it nests most commonly used density functions. In this paper, moments and maximum likelihood estimation of the parameters of ASGT distribution are given. Skewness and kurtosis measures are derived based on the first four noncentral moments. The cumulative distribution function (cdf) of ASGT distribution is also obtained. In the application part of the study, two real life problems taken from the literature are modeled by using ASGT distribution.</p>

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