scholarly journals Polynomial Interpolation of Normal Conditional Expectation

2019 ◽  
Vol 8 (3) ◽  
pp. 75
Author(s):  
Anis Rezgui
Keyword(s):  
Error Estimation ◽  
Conditional Expectation ◽  
Polynomial Function ◽  
Polynomial Interpolation ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Degree Polynomial ◽  
Reasonable Condition ◽  
Pointwise Error ◽  
Standard Normal

In this paper we are interested in approximating the conditional expectation of a given random variable X with respect to the standard normal distribution N(0, 1). Actually we have shown that the conditional expectation E(X|Z) could be interpolated by an N degree polynomial function of Z, φN(Z) where N is the number of observations recorded for the conditional expectation E(X|Z = z). A pointwise error estimation has been proved under reasonable condition on the random variable X.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Standard Normal ◽  
Correction Terms ◽  
Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

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 Central limit theorems for the real eigenvalues of large Gaussian random matrices

2017 ◽  
Vol 06 (01) ◽  
pp. 1750002 ◽  
Author(s):  
N. J. Simm
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Polynomial Function ◽  
Random Variable ◽  
Real Matrix ◽  
Two Component System ◽  
The Real ◽  
Two Component ◽  
Standard Normal ◽  
Independent Identically Distributed

Let [Formula: see text] be an [Formula: see text] real matrix whose entries are independent identically distributed standard normal random variables [Formula: see text]. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability 26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if [Formula: see text] are the real eigenvalues of [Formula: see text], then for any even polynomial function [Formula: see text] and even [Formula: see text], we have the convergence in distribution to a normal random variable [Formula: see text] as [Formula: see text], where [Formula: see text].

scholarly journals A Berry-Esseen bound for the lightbulb process

2011 ◽  
Vol 43 (03) ◽  
pp. 875-898 ◽  
Author(s):  
Larry Goldstein ◽  
Haimeng Zhang
Keyword(s):  
Conditional Expectation ◽  
Spectral Decomposition ◽  
Random Variable ◽  
Stein’S Method ◽  
Normal Random Variable ◽  
The Past ◽  
Terminal Time ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Size Bias

In the so-called lightbulb process, on daysr= 1,…,n, out ofnlightbulbs, all initially off, exactlyrbulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. WithXthe number of bulbs on at the terminal timen, an even integer, and μ =n/2, σ2= var(X), we have supz∈R| P((X- μ)/σ ≤z) - P(Z≤z) | ≤nΔ̅0/2σ2+ 1.64n/σ3+ 2/σ, whereZis a standard normal random variable and Δ̅0= 1/2√n+ 1/2n+ e−n/2/3 forn≥ 6, yielding a bound of orderO(n−1/2) asn→ ∞. A similar, though slightly larger bound, holds for oddn. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for evenndepends on the construction of a variableXson the same space asXthat has theX-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuousg, and for which there exists aB≥ 0, in this caseB= 2, such thatX≤Xs≤X+Balmost surely. The argument for oddnis similar to that for evenn, but one first couplesXclosely toV, a symmetrized version ofX, for which a size bias coupling ofVtoVscan proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

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.

scholarly journals A Berry-Esseen bound for the lightbulb process

2011 ◽  
Vol 43 (3) ◽  
pp. 875-898 ◽  
Author(s):  
Larry Goldstein ◽  
Haimeng Zhang
Keyword(s):  
Conditional Expectation ◽  
Spectral Decomposition ◽  
Random Variable ◽  
Stein’S Method ◽  
Normal Random Variable ◽  
The Past ◽  
Terminal Time ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Size Bias

In the so-called lightbulb process, on days r = 1,…,n, out of n lightbulbs, all initially off, exactly r bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With X the number of bulbs on at the terminal time n, an even integer, and μ = n/2, σ2 = var(X), we have supz ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 2/σ, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n−1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even n depends on the construction of a variable Xs on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ Xs ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to Vs can proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

scholarly journals Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 77
Author(s):  
Juan Carlos Castro-Palacio ◽  
J. M. Isidro ◽  
Esperanza Navarro-Pardo ◽  
Luisberis Velázquez-Abad ◽  
Pedro Fernández-de-Córdoba
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Discrete Model ◽  
Response Times ◽  
Random Variable ◽  
Ideal Gas ◽  
Standard Normal Distribution ◽  
Square Root ◽  
Unequal Variances ◽  
Standard Normal

The Chi distribution is a continuous probability distribution of a random variable obtained from the positive square root of the sum of k squared variables, each coming from a standard Normal distribution (mean = 0 and variance = 1). The variable k indicates the degrees of freedom. The usual expression for the Chi distribution can be generalised to include a parameter which is the variance (which can take any value) of the generating Gaussians. For instance, for k = 3, we have the case of the Maxwell-Boltzmann (MB) distribution of the particle velocities in the Ideal Gas model of Physics. In this work, we analyse the case of unequal variances in the generating Gaussians whose distribution we will still represent approximately in terms of a Chi distribution. We perform a Monte Carlo simulation to generate a random variable which is obtained from the positive square root of the sum of k squared variables, but this time coming from non-standard Normal distributions, where the variances can take any positive value. Then, we determine the boundaries of what to expect when we start from a set of unequal variances in the generating Gaussians. In the second part of the article, we present a discrete model to calculate the parameter of the Chi distribution in an approximate way for this case (unequal variances). We also comment on the application of this simple discrete model to calculate the parameter of the MB distribution (Chi of k = 3) when it is used to represent the reaction times to visual stimuli of a collective of individuals in the framework of a Physics inspired model we have published in a previous work.

scholarly journals Mathematical Conjecture for Random Complex Distributions of Random Complex Variables and their Common Statistics

2016 ◽  
Vol 1 (1) ◽  
pp. 10
Author(s):  
Yiyan Chen
Keyword(s):  
Complex Variable ◽  
Joint Distribution ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Paper Study ◽  
Two Dimensional ◽  
Font Size ◽  
Standard Normal ◽  
Complex Variance ◽  
Random Complex

<p><span style="font-family: Times New Roman; font-size: medium;">The random complex variable is a kind of random variable which in complex number as the random variable. In this paper, study on mathematical conjecture of random complex variable, two-dimensional joint distribution and complex expected, complex variance and other common statistics in the real and imaginary part of the random variables satisfy the independent standard normal distribution condition.</span></p>

Sebaran Peluang Acak Kontinu, Distribusi Normal, Distribusi Normal Baku, Distribusi T, Distribusi Chi Square, dan Distribusi F

2020 ◽  
Author(s):  
Ahmad Sudi Pratikno
Keyword(s):  
Hypothesis Test ◽  
Standard Normal Distribution ◽  
Process Data ◽  
Nominal Data ◽  
Chi Square ◽  
Stable Curve ◽  
Standard Normal ◽  
Research Findings ◽  
F Distribution ◽  
T Distribution

In statistics, there are various terms that may feel unfamiliar to researcher who is not accustomed to discussing it. However, despite all of many functions and benefits that we can get as researchers to process data, it will later be interpreted into a conclusion. And then researcher can digest and understand the research findings. The distribution of continuous random opportunities illustrates obtaining opportunities with some detection of time, weather, and other data obtained from the field. The standard normal distribution represents a stable curve with zero mean and standard deviation 1, while the t distribution is used as a statistical test in the hypothesis test. Chi square deals with the comparative test on two variables with a nominal data scale, while the f distribution is often used in the ANOVA test and regression analysis.

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