scholarly journals Probabilities of Competing Binomial Random Variables

2012 ◽  
Vol 49 (3) ◽  
pp. 731-744
Author(s):  
Wenbo V. Li ◽  
Vladislav V. Vysotsky
Keyword(s):  
Phase Transition ◽  
Fourier Analysis ◽  
Random Variables ◽  
Integral Representations ◽  
Transition Phenomenon ◽  
Convexity Properties ◽  
Binomial Random Variables

Suppose that both you and your friend toss an unfair coin n times, for which the probability of heads is equal to α. What is the probability that you obtain at least d more heads than your friend if you make r additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of n, and demonstrate surprising phase transition phenomenon as the parameters d, r, and α vary. Our main tools are integral representations based on Fourier analysis.

scholarly journals Probabilities of Competing Binomial Random Variables

2012 ◽  
Vol 49 (03) ◽  
pp. 731-744
Author(s):  
Wenbo V. Li ◽  
Vladislav V. Vysotsky
Keyword(s):  
Phase Transition ◽  
Fourier Analysis ◽  
Random Variables ◽  
Integral Representations ◽  
Transition Phenomenon ◽  
Convexity Properties ◽  
Binomial Random Variables

Suppose that both you and your friend toss an unfair coinntimes, for which the probability of heads is equal to α. What is the probability that you obtain at leastdmore heads than your friend if you makeradditional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function ofn, and demonstrate surprising phase transition phenomenon as the parametersd,r, and α vary. Our main tools are integral representations based on Fourier analysis.

scholarly journals Phase transition of the 2-Choices dynamics on core–periphery networks

2021 ◽  
Author(s):  
Emilio Cruciani ◽  
Emanuele Natale ◽  
André Nusser ◽  
Giacomo Scornavacca
Keyword(s):  
Phase Transition ◽  
Voting Behavior ◽  
Structural Parameters ◽  
Connected Subset ◽  
Transition Phenomenon ◽  
The Core ◽  
Analytical Results ◽  
Heterogeneous Behavior

AbstractThe 2-Choices dynamics is a process that models voting behavior on networks and works as follows: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. We study its behavior on a class of networks with core–periphery structure. Assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. We prove that, depending on the strength of the cut between core and periphery, a phase-transition phenomenon occurs: Either the core’s opinion rapidly spreads across the network, or a metastability phase takes place in which both opinions coexist for superpolynomial time. The interest of our result, which we also validate with extensive experiments on real networks, is twofold. First, it sheds light on the influence of the core on the rest of the network as a function of its connectivity toward the latter. Second, it is one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network.

Ordering comparison of negative binomial random variables with their mixtures

2008 ◽  
Vol 78 (14) ◽  
pp. 2234-2239 ◽  
Author(s):  
Mohammad Hossein Alamatsaz ◽  
Somayyeh Abbasi
Keyword(s):  
Negative Binomial ◽  
Random Variables ◽  
Binomial Random Variables

Fourier Analysis and Benfordʼs Law

Benford's Law ◽  
2015 ◽  
Author(s):  
Steven J. Miller
Keyword(s):  
Distribution Function ◽  
Fourier Analysis ◽  
Random Matrix ◽  
Matrix Theory ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Benford’S Law ◽  
Good System ◽  
Benford's Law ◽  
Geometric Brownian Motions

This chapter continues the development of the theory of Benford's law. It uses Fourier analysis (in particular, Poisson Summation) to prove many systems either satisfy or almost satisfy the Fundamental Equivalence, and hence either obey Benford's law, or are well approximated by it. Examples range from geometric Brownian motions to random matrix theory to products and chains of random variables to special distributions. The chapter furthermore develops the notion of a Benford-good system. Unfortunately one of the conditions here concerns the cancelation in sums of translated errors related to the cumulative distribution function, and proving the required cancelation often requires techniques specific to the system of interest.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Exponential Random Variables ◽  
Binomial Random Variables

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

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