Dependencies Between Histogram Parameters and the Kernel Estimate of the Probability Density of a Multidimensional Random Variable

2020 ◽  
Vol 62 (11) ◽  
pp. 945-952
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Random Variable ◽  
Kernel Estimate

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 Linear behavior in Covid19 epidemic as an effect of lockdown

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Dario Bambusi ◽  
Antonio Ponno
Keyword(s):  
Probability Density ◽  
Linear Growth ◽  
Random Variable ◽  
Linear Decrease ◽  
Linear Behavior ◽  
The Law

AbstractWe propose a mechanism explaining the approximately linear growth of Covid19 world total cases as well as the slow linear decrease of the daily new cases (and daily deaths) observed (in average) in USA and Italy. In our explanation, we regard a given population (the whole world or a single nation) as composed by many sub-clusters which, after lockdown, evolve essentially independently. The interaction is modeled by the fact that the outbreak time of the epidemic in a sub-cluster is a random variable with probability density slowly varying in time. The explanation is independent of the law according to which the epidemic evolves in the single sub cluster.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

Reliability Methods for Bimodal Distribution With First-Order Approximation1

2018 ◽  
Vol 5 (1) ◽  
Author(s):  
Zhangli Hu ◽  
Xiaoping Du
Keyword(s):  
Probability Density ◽  
Order Approximation ◽  
Limit State ◽  
Random Variables ◽  
Random Variable ◽  
State Function ◽  
First Order ◽  
Reliability Method ◽  
Reliability Methods ◽  
Basic Random Variable

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. When binomial variables are involved, traditional reliability methods, such as the first-order second moment (FOSM) method and the first-order reliability method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation (SPA) for bimodal variables and then employs SPA-based reliability methods with first-order approximation to predict the reliability. A limit-state function is at first approximated with the first-order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The SPA is then applied to estimate the reliability. Examples show that the SPA-based reliability methods are more accurate than FOSM and FORM.

A generalized Ostrowski type inequality for a random variable whose probability density function belongs to L∞[a, b]

2008 ◽  
Vol 41 (3) ◽  
Author(s):  
Arif Rafiq ◽  
Nazir Ahmad Mir ◽  
Fiza Zafar
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Type Inequality ◽  
Random Variable ◽  
Ostrowski Type Inequality

AbstractWe establish here an inequality of Ostrowski type for a random variable whose probability density function belongs to L

scholarly journals Determining the First Probability Density Function of Linear Random Initial Value Problems by the Random Variable Transformation (RVT) Technique: A Comprehensive Study

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
M.-C. Casabán ◽  
J.-C. Cortés ◽  
J.-V. Romero ◽  
M.-D. Roselló
Keyword(s):  
Stochastic Process ◽  
Probability Density Function ◽  
Differential Equations ◽  
Probability Density ◽  
Density Function ◽  
Initial Value Problems ◽  
Random Variable ◽  
Initial Value ◽  
Variable Transformation ◽  
Comprehensive Study

Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.

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.   

scholarly journals A nonparametric algorithm for automatic classification of large multivariate statistical data sets and its application

2021 ◽  
Vol 45 (2) ◽  
pp. 253-260
Author(s):  
I.V. Zenkov ◽  
A.V. Lapko ◽  
V.A. Lapko ◽  
S.T. Im ◽  
V.P. Tuboltsev ◽  
...  
Keyword(s):  
Probability Density ◽  
Random Variables ◽  
Random Variable ◽  
Automatic Classification ◽  
Optimal Number ◽  
Data Sets ◽  
Nonparametric Algorithm ◽  
Optimal Discretization ◽  
Sampling Intervals ◽  
Range Of Values

A nonparametric algorithm for automatic classification of large statistical data sets is proposed. The algorithm is based on a procedure for optimal discretization of the range of values of a random variable. A class is a compact group of observations of a random variable corresponding to a unimodal fragment of the probability density. The considered algorithm of automatic classification is based on the «compression» of the initial information based on the decomposition of a multidimensional space of attributes. As a result, a large statistical sample is transformed into a data array composed of the centers of multidimensional sampling intervals and the corresponding frequencies of random variables. To substantiate the optimal discretization procedure, we use the results of a study of the asymptotic properties of a kernel-type regression estimate of the probability density. An optimal number of sampling intervals for the range of values of one- and two-dimensional random variables is determined from the condition of the minimum root-mean square deviation of the regression probability density estimate. The results obtained are generalized to the discretization of the range of values of a multidimensional random variable. The optimal discretization formula contains a component that is characterized by a nonlinear functional of the probability density. An analytical dependence of the detected component on the antikurtosis coefficient of a one-dimensional random variable is established. For independent components of a multidimensional random variable, a methodology is developed for calculating estimates of the optimal number of sampling intervals for random variables and their lengths. On this basis, a nonparametric algorithm for the automatic classification is developed. It is based on a sequential procedure for checking the proximity of the centers of multidimensional sampling intervals and relationships between frequencies of the membership of the random variables from the original sample of these intervals. To further increase the computational efficiency of the proposed automatic classification algorithm, a multithreaded method of its software implementation is used. The practical significance of the developed algorithms is confirmed by the results of their application in processing remote sensing data.

scholarly journals Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

2017 ◽  
Vol 2 (1) ◽  
pp. 213-224 ◽  
Author(s):  
J.-C. Cortés ◽  
A. Navarro-Quiles ◽  
J.-V. Romero ◽  
M.-D. Roselló
Keyword(s):  
Differential Equation ◽  
Probability Density ◽  
Random Variable ◽  
Probability Density Functions ◽  
Point Of View ◽  
Density Functions ◽  
Statistical Point ◽  
Covariance Functions ◽  
Infinite Point ◽  
Homogeneous Linear

AbstractIn this paper the randomized Cauchy-Euler differential equation is studied. With this aim, from a statistical point of view, both the first and second probability density functions of the solution stochastic process are computed. Then, the main statistical functions, namely, the mean, the variance and the covariance functions are determined as well. The study includes the computation of the first and second probability density functions of the regular-singular infinite point via an adequate mapping transforming the problem about the origin. The study is strongly based upon the Random Variable Transformation technique along with some results that have been recently published by some of authors to the random homogeneous linear second-order differential equation. Finally, an illustrative example is shown.

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