Probability measures on fractal curves (probability distributions on the Vicsek fractal)

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Mykola V. Pratsiovytyi ◽  
Valeriy M. Kovalenko
Keyword(s):  
Probability Distributions ◽  
Random Variable ◽  
Probability Measures ◽  
Fractal Properties ◽  
Complex Valued

AbstractIn this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable ζ = ∑

scholarly journals Dual Lukacs regressions of negative orders for noncommutative variables

2014 ◽  
Vol 17 (03) ◽  
pp. 1450021 ◽  
Author(s):  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Random Variable ◽  
Probability Measures ◽  
Classical Probability ◽  
Binomial Distributions ◽  
Beta Distributions

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable 𝕍1/2(𝕀 - 𝕌)𝕍1/2 given by 𝕍1/2𝕌𝕍1/2, where 𝕌 and 𝕍 are free, we characterize free Poisson and free binomial distributions. Our paper is a free probability analogue of results known in classical probability,3 where gamma and beta distributions are characterized by constancy of 𝔼((V(1 - U))i|UV), for i ∈ {-2, -1, 1, 2}. This paper together with previous results18 exhaust all cases of characterizations from Ref. 3.

scholarly journals Lattice Paths, Sampling Without Replacement, and Limiting Distributions

10.37236/156 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer ◽  
H. Prodinger
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Random Variable ◽  
Phase Changes ◽  
Lattice Paths ◽  
Sampling Without Replacement ◽  
Sample Paths ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

Introduction to Information Theory

2014 ◽  
Author(s):  
M. Vidyasagar
Keyword(s):  
Information Theory ◽  
Probability Distribution ◽  
Relative Entropy ◽  
Probability Distributions ◽  
Random Variables ◽  
Conditional Entropy ◽  
Random Variable ◽  
Entropy Function ◽  
Concave Functions ◽  
Leibler Divergence

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random variable (or more precisely, the probability distribution of the random variable). When there are two or more random variables, it is worthwhile to study the conditional entropy of one random variable with respect to another. The last concept is relative entropy, also known as the Kullback–Leibler divergence, which measures the “disparity” between two probability distributions. The chapter first considers convex and concave functions before discussing the properties of the entropy function, conditional entropy, uniqueness of the entropy function, and the Kullback–Leibler divergence.

scholarly journals Higher-Order Cumulants Drive Neuronal Activity Patterns, Inducing UP-DOWN States in Neural Populations

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 477 ◽  
Author(s):  
Roman Baravalle ◽  
Fernando Montani
Keyword(s):  
Probability Distributions ◽  
Activity Patterns ◽  
Gaussian Model ◽  
Random Variable ◽  
Neural System ◽  
Higher Order ◽  
Second Order ◽  
Specific Information ◽  
Neuronal Populations ◽  
Binder Cumulant

A major challenge in neuroscience is to understand the role of the higher-order correlations structure of neuronal populations. The dichotomized Gaussian model (DG) generates spike trains by means of thresholding a multivariate Gaussian random variable. The DG inputs are Gaussian distributed, and thus have no interactions beyond the second order in their inputs; however, they can induce higher-order correlations in the outputs. We propose a combination of analytical and numerical techniques to estimate higher-order, above the second, cumulants of the firing probability distributions. Our findings show that a large amount of pairwise interactions in the inputs can induce the system into two possible regimes, one with low activity (“DOWN state”) and another one with high activity (“UP state”), and the appearance of these states is due to a combination between the third- and fourth-order cumulant. This could be part of a mechanism that would help the neural code to upgrade specific information about the stimuli, motivating us to examine the behavior of the critical fluctuations through the Binder cumulant close to the critical point. We show, using the Binder cumulant, that higher-order correlations in the outputs generate a critical neural system that portrays a second-order phase transition.

Characterization of matrix probability distributions by mean residual lifetime

1986 ◽  
Vol 100 (3) ◽  
pp. 583-589
Author(s):  
P. E. Jupp
Keyword(s):  
Symmetric Matrix ◽  
Probability Distributions ◽  
Random Variable ◽  
Residual Lifetime ◽  
Regularity Conditions ◽  
Mean Residual Lifetime ◽  
The Mean ◽  
Image Position ◽  
Vector Valued

The mean residual lifetime of a real-valued random variable X is the function e defined byOne of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz [10]. References to related characterizations are given by Galambos and Kotz [3], pages 30–35. It was established by Jupp and Mardia[6] that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.

Multivariate monotone likelihood ratio and uniform conditional stochastic order

1982 ◽  
Vol 19 (3) ◽  
pp. 695-701 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Conditional Probability ◽  
Likelihood Ratio ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Total Positivity ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Fkg Inequality ◽  
The Fkg Inequality

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

Modeling Uncertainties in Power System by Generalized Lambda Distribution

2014 ◽  
Vol 15 (3) ◽  
pp. 195-203 ◽  
Author(s):  
Qing Xiao
Keyword(s):  
Power System ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Polynomial Equation ◽  
Generalized Lambda Distribution ◽  
Modeling Uncertainties ◽  
Probability Weighted Moment

Abstract This paper employs the generalized lambda distribution (GLD) to model random variables with various probability distributions in power system. In the context of the probability weighted moment (PWM), an optimization-free method is developed to assess the parameters of GLD. By equating the first four PWMs of GLD with those of the target random variable, a polynomial equation with one unknown is derived to solve for the parameters of GLD. When employing GLD to model correlated multivariate random variables, a method of accommodating the dependency is put forward. Finally, three examples are worked to demonstrate the proposed method.

scholarly journals Characterizations of Some Probability Distributions with Completely Monotonic Density Functions

2021 ◽  
pp. 51-64
Author(s):  
Mohammad Shakil ◽  
Dr. Mohammad Ahsanullah ◽  
Dr. B. M. G. Kibria Kibria
Keyword(s):  
Probability Distribution ◽  
Probability Density ◽  
Probability Distributions ◽  
Random Variable ◽  
Record Values ◽  
Density Functions ◽  
Exponential Distributions ◽  
Completely Monotonic ◽  
Percentage Points ◽  
Generalized Gamma Function

For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.   

A CHARACTERIZATION OF PROBABILITY MEASURES IN TERMS OF WICK PRODUCT INEQUALITIES

2008 ◽  
Vol 11 (03) ◽  
pp. 377-391 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Random Variable ◽  
Wick Product ◽  
Probability Measures ◽  
Polynomial Functions ◽  
Finite Moments ◽  
Normally Distributed

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds: [Formula: see text] We show that this result can be extended to a random variable X, not necessary Gaussian, having an infinite support and finite moments of all orders, if and only if its Szegö–Jacobi sequence {ωk}k ≥ 1 is super-additive.

Sign in / Sign up

Export Citation Format

Share Document