Cumulative Distribution Function for the Standard Normal Random Variable

1989 ◽  
pp. 55-56
Author(s):  
Stephen Kokoska ◽  
Christopher Nevison
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Normal Random Variable ◽  
Cumulative Distribution ◽  
Standard Normal Random Variable ◽  
Standard Normal

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 On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
Author(s):  
Jalal Chachi
Keyword(s):  
Probability Theory ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Distribution Characteristics ◽  
Fuzzy Random Variables ◽  
Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

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 Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

2013 ◽  
Vol 50 (4) ◽  
pp. 909-917
Author(s):  
M. Bondareva
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Step Function ◽  
Cumulative Distribution ◽  
Control Parameters ◽  
Poisson Random Variable ◽  
The Mean ◽  
Standard Deviations

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

scholarly journals Precise Rates in Log Laws for NA Sequences

2007 ◽  
Vol 2007 ◽  
pp. 1-11
Author(s):  
Yuexu Zhao
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Negatively Associated ◽  
Standard Normal Random Variable ◽  
Na Random Variables ◽  
Standard Normal ◽  
Na Sequences

LetX1,X2,…be a strictly stationary sequence of negatively associated (NA) random variables withEX1=0, setSn=X1+⋯+Xn, suppose thatσ2=EX12+2∑n=2∞EX1Xn>0andEX12<∞,if−1<α≤1;EX12(log|X1|)α<∞, ifα>1. We provelimε↓0ε2α+2∑n=1∞((logn)α/n)P(|Sn|≥σ(ε+κn)2nlogn)=2−(α+1)(α+1)−1E|N|2α+2, whereκn=O(1/logn)and N is the standard normal random variable.

ON CUMULATIVE RESIDUAL EXTROPY

2019 ◽  
Vol 34 (4) ◽  
pp. 605-625 ◽  
Author(s):  
S. M. A. Jahanshahi ◽  
H. Zarei ◽  
A. H. Khammar
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Alternative Measure ◽  
Numerical Examples ◽  
Cumulative Residual ◽  
Measure Of Uncertainty

Recently, an alternative measure of uncertainty called extropy is proposed by Lad et al. [12]. The extropy is a dual of entropy which has been considered by researchers. In this article, we introduce an alternative measure of uncertainty of random variable which we call it cumulative residual extropy. This measure is based on the cumulative distribution function F. Some properties of the proposed measure, such as its estimation and applications, are studied. Finally, some numerical examples for illustrating the theory are included.

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.

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 &rdquo;random number&rdquo; will be used along the text to mean &rdquo;pseudo-random number&rdquo;.

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