scholarly journals On the Bivariate Nakagami-m Cumulative Distribution Function: Closed-Form Expression and Applications

2013 ◽  
Vol 61 (4) ◽  
pp. 1404-1414 ◽  
Author(s):  
F. J. Lopez-Martinez ◽  
D. Morales-Jimenez ◽  
E. Martos-Naya ◽  
J. F. Paris
Keyword(s):  
Distribution Function ◽  
Closed Form ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals A NOTE CONCERNING THE DISTANCES OF UNIFORMLY DISTRIBUTED POINTS FROM THE CENTRE OF A RECTANGLE

2012 ◽  
Vol 87 (1) ◽  
pp. 115-119 ◽  
Author(s):  
ROBERT STEWART ◽  
HONG ZHANG
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Average Distance ◽  
Cumulative Distribution

AbstractGiven a rectangle containing uniformly distributed random points, how far are the points from the rectangle’s centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.

scholarly journals On the Computation of Integrals of Bivariate Gaussian Distribution

2020 ◽  
Author(s):  
Vincent Savaux ◽  
Luc Le Magoarou
Keyword(s):  
Closed Form ◽  
Gaussian Distribution ◽  
Cumulative Distribution ◽  
Closed Form Expression ◽  
Geometrical Approach ◽  
Trigonometric Functions ◽  
Double Integral ◽  
Form Expression ◽  
Angular Sector ◽  
Bivariate Gaussian Distribution

This paper deals with the computation of integrals<br>of centred bivariate Gaussian densities over any domain defined as an angular sector of R^2. Based on an accessible geometrical approach of the problem, we suggest to transform the double integral into a single one, leading to a tractable closed-form expression only involving trigonometric functions. This solution can also be seen as the angular cumulative distribution of bivariate centered Gaussian variables (X,Y). We aim to provide a didactic approach of our results, and we validate them by comparing with those of the literature.

scholarly journals On the Computation of Integrals of Bivariate Gaussian Distribution

2020 ◽  
Author(s):  
Vincent Savaux ◽  
Luc Le Magoarou
Keyword(s):  
Closed Form ◽  
Gaussian Distribution ◽  
Cumulative Distribution ◽  
Closed Form Expression ◽  
Geometrical Approach ◽  
Trigonometric Functions ◽  
Double Integral ◽  
Form Expression ◽  
Angular Sector ◽  
Bivariate Gaussian Distribution

This paper deals with the computation of integrals<br>of centred bivariate Gaussian densities over any domain defined as an angular sector of R^2. Based on an accessible geometrical approach of the problem, we suggest to transform the double integral into a single one, leading to a tractable closed-form expression only involving trigonometric functions. This solution can also be seen as the angular cumulative distribution of bivariate centered Gaussian variables (X,Y). We aim to provide a didactic approach of our results, and we validate them by comparing with those of the literature.

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