scholarly journals Sigma-Pi Structure with Bernoulli Random Variables: Power-Law Bounds for Probability Distributions and Growth Models with Interdependent Entities

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 241
Author(s):  
Arthur Matsuo Yamashita Rios de Sousa ◽  
Hideki Takayasu ◽  
Didier Sornette ◽  
Misako Takayasu
Keyword(s):  
Probability Distribution ◽  
Power Law ◽  
Extreme Values ◽  
Growth Models ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Coefficients ◽  
Bernoulli Random Variables ◽  
Product Term ◽  
Simple Growth

The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.

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 Estimation of the agricultural probability of loss: evidence for soybean in Paraná state

2014 ◽  
Vol 52 (1) ◽  
pp. 25-40
Author(s):  
Vitor Augusto Ozaki ◽  
Ricardo Olinda ◽  
Priscila Neves Faria ◽  
Rogério Costa Campos
Keyword(s):  
Probability Distribution ◽  
Extreme Values ◽  
Probability Distributions ◽  
Agricultural Insurance ◽  
Insurance Contracts ◽  
Normality Assumption ◽  
Insurance Program ◽  
Symmetrical Distribution ◽  
Family Of Distributions ◽  
Accurate Quantification

In any agricultural insurance program, the accurate quantification of the probability of the loss has great importance. In order to estimate this quantity, it is necessary to assume some parametric probability distribution. The objective of this work is to estimate the probability of loss using the theory of the extreme values modeling the left tail of the distribution. After that, the estimated values will be compared to the values estimated under the normality assumption. Finally, we discuss the implications of assuming a symmetrical distribution instead of a more flexible family of distributions when estimating the probability of loss and pricing the insurance contracts. Results show that, for the selected regions, the probability distributions present a relative degree of skewness. As a consequence, the probability of loss is quite different from those estimated supposing the Normal distribution, commonly used by Brazilian insurers.

scholarly journals Uniqueness of the Level Two Bayesian Network Representing a Probability Distribution

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Linda Smail
Keyword(s):  
Artificial Intelligence ◽  
Probability Distribution ◽  
Bayesian Networks ◽  
Bayesian Network ◽  
Probabilistic Models ◽  
Probability Distributions ◽  
Random Variables ◽  
Inference Problem ◽  
Computation Algorithm

Bayesian Networks are graphic probabilistic models through which we can acquire, capitalize on, and exploit knowledge. they are becoming an important tool for research and applications in artificial intelligence and many other fields in the last decade. This paper presents Bayesian networks and discusses the inference problem in such models. It proposes a statement of the problem and the proposed method to compute probability distributions. It also uses D-separation for simplifying the computation of probabilities in Bayesian networks. Given a Bayesian network over a family of random variables, this paper presents a result on the computation of the probability distribution of a subset of using separately a computation algorithm and D-separation properties. It also shows the uniqueness of the obtained result.

scholarly journals Asymptotic expansions for distributions of sums of a double sequence of quasilattice random variables

2011 ◽  
Vol 52 ◽  
pp. 353-358
Author(s):  
Algimantas Bikelis ◽  
Juozas Augutis ◽  
Kazimieras Padvelskis
Keyword(s):  
Asymptotic Expansion ◽  
Probability Distribution ◽  
Asymptotic Expansions ◽  
Probability Distributions ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible ◽  
Formal Asymptotic Expansion

We consider the formal asymptotic expansion of probability distribution of the sums of independent random variables. The approximation was made by using infinitely divisible probability distributions.  

scholarly journals Max Weibull-G Power Series Distributions

2021 ◽  
pp. 15-29
Author(s):  
Munteanu Bogdan Gheorghe
Keyword(s):  
Probability Distribution ◽  
Power Series ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
New Family ◽  
The Family ◽  
Distribution Family ◽  
Power Series Distributions

Based on the Weibull-G Power probability distribution family, we have proposed a new family of probability distributions, named by us the Max Weibull-G power series distributions, which may be applied in order to solve some reliability problems. This implies the fact that the Max Weibull-G power series is the distribution of a random variable max (X1 ,X2 ,...XN) where X1 ,X2 ,... are Weibull-G distributed independent random variables and N is a natural random variable the distribution of which belongs to the family of power series distribution. The main characteristics and properties of this distribution are analyzed.

The Effect of Corrosion Growth Model Assumptions on the Reliability Estimates of Corroded Pipelines

2016 ◽  
Author(s):  
Markus R. Dann ◽  
Luc Huyse
Keyword(s):  
Stochastic Process ◽  
Corrosion Rate ◽  
Probabilistic Models ◽  
Growth Models ◽  
Probability Distributions ◽  
Random Variables ◽  
Theoretical Background ◽  
Process Models ◽  
Metal Loss ◽  
Reliability Estimates

Corrosion growth models are used to estimate future metal loss and the safe remaining lifetime of corrosion features in pipelines. Probabilistic models have become increasingly important in practice for reliability and risk-based pipeline assessments. The unknown model variables are usually determined from in-line inspection (ILI) results. Corrosion growth models exhibit various levels of complexity to account for temporal and spatial uncertainties of the actual corrosion growth process, and measurement uncertainties associated with ILIs. Model diversity leads to significant differences in how the models approach the uncertainty of future corrosion growth. This paper builds upon previous work and provides some theoretical background to an application described in [1]. It compares four common probabilistic corrosion growth models with respect to reliability estimates of leak failure. The four models are two uncertain corrosion rate models and two stochastic process models where the features are considered to be either independent or exchangeable. The unknown random variables of each model are updated in a Bayesian manner using the same ILI results. The key findings of this paper are: • Proper truncation at zero of the probability distributions for the unknown random variables is necessary if the measured corrosion growth is near zero or negative. • A stochastic process leads to lower uncertainties when determining future metal loss and, consequently, an increased reliability against leak failure than corrosion rate models. • The assumption of exchangeable features causes a reduction in the probability of leak failure due to the effect of borrowing information compared to independent features. The four corrosion growth models provide similar results with respect to the probability of failure if the measured corrosion growth is large. As the measured corrosion growth decreases in size, the differences between the reliability estimates increase.

scholarly journals Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT

2019 ◽  
Author(s):  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Probability Distribution ◽  
Power Law ◽  
General Model ◽  
Probability Distributions ◽  
Step Function ◽  
Propositional Satisfiability ◽  
Wide Range ◽  
Random Instances ◽  
Variable Probability

We study a more general model to generate random instances of Propositional Satisfiability (SAT) with n Boolean variables, m clauses, and exactly k variables per clause. Additionally, our model is given an arbitrary probability distribution (p_1, ..., p_n) on the variable occurrences. Therefore, we call it non-uniform random k-SAT. The number m of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We identify conditions on the variable probability distribution (p_1, ..., p_n) under which the satisfiability threshold is sharp if its position is already known asymptotically. This result generalizes Friedgut’s sharpness result from uniform to non-uniform random k -SAT and implies sharpness for thresholds of a wide range of random k -SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law.

INTENSITY PROBABILITY DISTRIBUTION OF SPIKING-MODE OSCILLATIONS IN LASERS: POWER-LAW UNIVERSALITY

1994 ◽  
Vol 04 (01) ◽  
pp. 225-230
Author(s):  
FUMIHIKO ISHIYAMA ◽  
KENJU OTSUKA
Keyword(s):  
Probability Distribution ◽  
Power Law ◽  
Probability Distributions ◽  
Delayed Feedback ◽  
Output Intensity ◽  
External Perturbations ◽  
Loss Modulation

We theoretically investigated the probability distributions of the output intensity of lasers with external perturbations. These perturbations were loss modulation and incoherent delayed feedback. We found that the intensity probability distributions usually obey a distinct power-law in spiking-mode oscillations both in periodic and chaotic regimes.

A note on the Probability that first component fails before the second component

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Ehtesham Husain ◽  
Masood ul Haq
Keyword(s):  
Probability Distribution ◽  
Failure Time ◽  
Random Variables ◽  
Life Time ◽  
Life Testing ◽  
Human Organs

<p><span>The reliability (unreliability) and life testing are important topics in the field of engineering, electronic, <span>medicine, economic and many more, where we are interested in, life of components, human organs, <span>subsystem and system. Statistically, a probability distribution failure time (life time) of a certain form is <span>usually assumed to give reliability of a component for a system for each time t. Some well known <span>parametric life time models (T ≥ 0) are Exponential, Weibull, Inverse Weibull, Gamma, Lognormal, <span>normal ( T&gt;0 ; left truncated ) etc. </span></span></span></span></span></span></p><p><span><span><span><span><span><span><span>In this paper we consider a system that, has two components with independent but non-identical life time <span>probabilities explained by two distinct random variables say T<span>1 <span>and T<span>2 <span>, where T<span>1 <span>has a constant hazard <span>rate and T<span>2 <span>has an increasing hazard respectively </span></span></span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></span></span></span></span></span></span></span></span></p>

POWER LAW AND STRETCHED EXPONENTIAL EFFECTS OF EXTREME EVENTS IN CHINESE STOCK MARKETS

2010 ◽  
Vol 09 (02) ◽  
pp. 203-217 ◽  
Author(s):  
XIAOJUN ZHAO ◽  
PENGJIAN SHANG ◽  
YULEI PANG
Keyword(s):  
Correlation Analysis ◽  
Power Law ◽  
Extreme Events ◽  
Stock Markets ◽  
Cross Correlation ◽  
Extreme Values ◽  
Cumulative Distribution ◽  
Stretched Exponential ◽  
Cross Correlation Analysis ◽  
Chinese Stock Markets

This paper reports the statistics of extreme values and positions of extreme events in Chinese stock markets. An extreme event is defined as the event exceeding a certain threshold of normalized logarithmic return. Extreme values follow a piecewise function or a power law distribution determined by the threshold due to a crossover. Extreme positions are studied by return intervals of extreme events, and it is found that return intervals yield a stretched exponential function. According to correlation analysis, extreme values and return intervals are weakly correlated and the correlation decreases with increasing threshold. No long-term cross-correlation exists by using the detrended cross-correlation analysis (DCCA) method. We successfully introduce a modification specific to the correlation and derive the joint cumulative distribution of extreme values and return intervals at 95% confidence level.

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