A Note On The Calculation And Application Of The Third Absolute Central Moment Of Chi-square Distribution

1992 ◽  
pp. 83-86
Author(s):  
Abu Hassan Shaari Mohd Nor ◽  
Fauziah Maarof
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variable ◽  
Central Moment ◽  
Chi Square ◽  
Exact Bound ◽  
The Third ◽  
The Central Limit Theorem ◽  
Numerical Integrations

Kertas ini mengemukakan satu cara mencari momen memusat mutlak ketiga bagi pembolehubah rawak khi-kuasadua. Aturcara SAS (1988) iaitu PROBCHI digunakan bagi menyelesaikan pengamiran berangka. Kegunaannya dalam membina batas yang tepat ke atas ralat penghampiran dalam Teorem Had Memusat diberikan. This paper presents a way of calculating the third absolute central moment of a chi-square random variable. The SAS (1988) function PROBCHI is used to evaluate the numerical integrations. An application of this result in the construction of an exact bound on the error of approximation in the Central Limit Theorem is presented.

Central Limit Theorems and Asymptotic Spectral Analysis on Large Graphs

1998 ◽  
Vol 01 (02) ◽  
pp. 221-246 ◽  
Author(s):  
Akihito Hora
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variable ◽  
Spectral Structure ◽  
Regular Graphs ◽  
Large Graphs ◽  
The Central Limit Theorem ◽  
Distance Regular Graphs

Regarding the adjacency matrix of a graph as a random variable in the framework of algebraic or noncommutative probability, we discuss a central limit theorem in which the size of a graph grows in several patterns. Various limit distributions are observed for some Cayley graphs and some distance-regular graphs. To obtain the central limit theorem of this type, we make combinatorial analysis of mixed moments of noncommutative random variables on one hand, and asymptotic analysis of spectral structure of the graph on the other hand.

scholarly journals Simulating the Central Limit Theorem

2018 ◽  
Vol 18 (2) ◽  
pp. 345-356
Author(s):  
Marshall A. Taylor
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Graduate Students ◽  
Central Limit ◽  
Standard Error ◽  
Random Variable ◽  
Sampling Distribution ◽  
The Central Limit Theorem ◽  
Variable Distribution

Understanding the central limit theorem is crucial for comprehending parametric inferential statistics. Despite this, undergraduate and graduate students alike often struggle with grasping how the theorem works and why researchers rely on its properties to draw inferences from a single unbiased random sample. In this article, I outline a new command, sdist, that can be used to simulate the central limit theorem by generating a matrix of randomly generated normal or nonnormal variables and comparing the true sampling distribution standard deviation with the standard error from the first randomly generated sample. The user also has the option of plotting the empirical sampling distribution of sample means, the first random variable distribution, and a stacked visualization of the two distributions.

scholarly journals Multiplexing: B

2021 ◽  
pp. 59-69
Author(s):  
Jean Walrand
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Characteristic Function ◽  
Central Limit ◽  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wifi Networks ◽  
The Central Limit Theorem ◽  
Communication Links

AbstractChapter 10.1007/978-3-030-49995-2_3 used the Central Limit Theorem to determine the number of users that can safely share a common cable or link. We saw that this result is also fundamental to calculate confidence intervals. In this section, we prove this theorem. A key tool is the characteristic function that provides a simple way to study sums of independent random variables.Section 4.1 introduces the characteristic function and calculates it for a Gaussian random variable. Section 4.2 uses that function to prove the Central Limit Theorem. Section 4.3 uses the characteristic function to calculate the moments of a Gaussian random variable. The sum of squares of Gaussian random variables is a common model of noise in communication links. Section 4.4 proves a remarkable property of such a sum. Section 4.5 shows how to use characteristic functions to approximate binomial and geometric random variables. The error function arises in the calculation of the probability of errors in transmission systems and also in decisions based on random observations. Section 4.6 derives useful approximations of that function. Section 4.7 concludes the chapter with a discussion of an adaptive multiple access protocol similar to one used in WiFi networks.

The Central Limit Theorem and the Law of Large Numbers for Pair-Connectivity in Bernoulli Trees

1989 ◽  
Vol 3 (4) ◽  
pp. 477-491
Author(s):  
Kyle T. Siegrist ◽  
Ashok T. Amin ◽  
Peter J. Slater
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Network Reliability ◽  
Random Variable ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Mean ◽  
The Law

Consider the standard network reliability model in which each edge of a given (n, m)-graph G is deleted, independently of all others, with probability q = 1– p (0 <p < 1). The pair-connectivity random variable X is defined to be the number of connected pairs of vertices that remain in G. The mean of X has been proposed as a measure of reliability for failure-prone communications networks in which the edge deletions correspond to failures of the communications links. We consider deviations from the mean, the law of large numbers, and the central limit theorem for X as n → ∞. Some explicit results are obtained when G is a tree using martingale difference sequences. Stars and paths are treated in detail.

scholarly journals Best predictors in logarithmic distance between positive random variables

2019 ◽  
Vol 15 (2) ◽  
pp. 15-28
Author(s):  
H. Gzyl
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Geodesic Distance ◽  
Geometric Mean ◽  
Random Variables ◽  
Random Variable ◽  
Metric Properties ◽  
The Central Limit Theorem ◽  
Logarithmic Distance

Abstract The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L1 norm. Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝn in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean. That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.

scholarly journals Simulating the Central Limit Theorem

2018 ◽  
Author(s):  
Marshall A. Taylor
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Graduate Students ◽  
Central Limit ◽  
Standard Error ◽  
Random Variable ◽  
Sampling Distribution ◽  
The Central Limit Theorem ◽  
Variable Distribution

Understanding the central limit theorem is crucial for comprehending parametric inferential statistics. Despite this, undergraduate and graduate students alike often struggle with grasping how the theorem works and why researchers rely on its properties to draw inferences from a single unbiased random sample. In this paper, I outline a new Stata package, sdist, which can be used to simulate the central limit theorem by generating a matrix of randomly generated normal or non-normal variables and comparing the true sampling distribution standard deviation to the standard error from the first randomly-generated sample. The user also has the option of plotting the empirical sampling distribution of sample means, the first random variable distribution, and a stacked visualization of the two distributions.

scholarly journals On the rate of convergence of moments in the central limit theorem

1978 ◽  
Vol 25 (2) ◽  
pp. 250-256 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Expansions ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
The Third ◽  
The Central Limit Theorem ◽  
Convergence Of Moments ◽  
Necessary And Sufficient

AbstractAn early extension of Lindeberg's central limit theorem was Bernstein's (1939) discovery of necessary and sufficient conditions for the convergence of moments in the central limit theorem. Von Bahr (1965) made a study of some asymptotic expansions in the central limit theorem, and obtained rates of convergence for moments. However, his results do not in general imply that the moments converge. Some better rates have been obtained by Bhattacharya and Rao for moments between the second and third. In this paper we give improved rates of convergence for absolute moments between the third and fourth.

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