scholarly journals A Note on Distributions in the Second Chaos

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1487
Author(s):  
Pauliina Ilmonen ◽  
Lauri Viitasaari
Keyword(s):  
Density Function ◽  
Random Variables ◽  
Basic Properties ◽  
Smooth Density

In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X = ∑ k ≥ 1 λ k ( ξ k 2 − 1 ) , where ξ k ∼ N ( 0 , 1 ) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λ k ≠ 0 infinitely often. Our work generalises some known results presented in the literature.

scholarly journals A numerical study of entropy and residual entropy estimators based on smooth density estimators for non-negative random variables

2021 ◽  
Vol 54 (2) ◽  
pp. 99-121
Author(s):  
Yogendra P. Chaubey ◽  
Nhat Linh Vu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Numerical Study ◽  
Random Variables ◽  
Random Variable ◽  
Residual Entropy ◽  
Density Estimator ◽  
Wide Range ◽  
Smooth Density

In this paper, we are interested in estimating the entropy of a non-negative random variable. Since the underlying probability density function is unknown, we propose the use of the Poisson smoothed histogram density estimator to estimate the entropy. To study the per- formance of our estimator, we run simulations on a wide range of densities and compare our entropy estimators with the existing estimators based on different approaches such as spacing estimators. Furthermore, we extend our study to residual entropy estimators which is the entropy of a random variable given that it has been survived up to time $t$.

Multivariate probabilistic rock physics models using Kumaraswamy distributions

Geophysics ◽  
2021 ◽  
pp. 1-43
Author(s):  
Dario Grana
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Random Variables ◽  
Rock Physics ◽  
Petrophysical Properties ◽  
Kumaraswamy Distribution ◽  
Model Predictions ◽  
Physics Model ◽  
Rock Physics Model

Rock physics models are physical equations that map petrophysical properties into geophysical variables, such as elastic properties and density. These equations are generally used in quantitative log and seismic interpretation to estimate the properties of interest from measured well logs and seismic data. Such models are generally calibrated using core samples and well log data and result in accurate predictions of the unknown properties. Because the input data are often affected by measurement errors, the model predictions are often uncertain. Instead of applying rock physics models to deterministic measurements, I propose to apply the models to the probability density function of the measurements. This approach has been previously adopted in literature using Gaussian distributions, but for petrophysical properties of porous rocks, such as volumetric fractions of solid and fluid components, the standard probabilistic formulation based on Gaussian assumptions is not applicable due to the bounded nature of the properties, the multimodality, and the non-symmetric behavior. The proposed approach is based on the Kumaraswamy probability density function for continuous random variables, which allows modeling double bounded non-symmetric distributions and is analytically tractable, unlike the Beta or Dirichtlet distributions. I present a probabilistic rock physics model applied to double bounded continuous random variables distributed according to a Kumaraswamy distribution and derive the analytical solution of the posterior distribution of the rock physics model predictions. The method is illustrated for three rock physics models: Raymer’s equation, Dvorkin’s stiff sand model, and Kuster-Toksoz inclusion model.

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.

Response Probability Analysis of Random Acoustic Field Based on Perturbation Stochastic Method and Change-of-Variable Technique

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Baizhan Xia ◽  
Dejie Yu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Acoustic Field ◽  
Random Variables ◽  
Stochastic Finite Element Method ◽  
Variable Technique ◽  
Change Of Variable

To calculate the probability density function of the response of a random acoustic field, a change-of-variable perturbation stochastic finite element method (CVPSFEM), which integrates the perturbation stochastic finite element method (PSFEM) and the change-of-variable technique in a unified form, is proposed. In the proposed method, the response of a random acoustic field is approximated as a linear function of the random variables based on a first order stochastic perturbation analysis. According to the linear relationship between the response and the random variables, the formal expression of the probability density function of the response of a random acoustic field is obtained by the change-of-variable technique. The numerical examples on a two-dimensional (2D) acoustic tube and a three-dimensional (3D) acoustic cavity of an automobile cabin verify the accuracy and efficiency of the proposed method. Hence, the proposed method can be considered as an effective method to quantify the effects of the parametric randomness of a random acoustic field on the sound pressure response.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (4) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

SummaryIn a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

scholarly journals Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

2009 ◽  
Vol 2009 ◽  
pp. 1-28 ◽  
Author(s):  
Xin Gao ◽  
Hong Xu ◽  
Dong Ye
Keyword(s):  
Asymptotic Behavior ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Random Variables ◽  
Tail Probability ◽  
Approximation Methods ◽  
Left And Right

We consider the asymptotic behavior of a probability density function for the sum of any two lognormally distributed random variables that are nontrivially correlated. We show that both the left and right tails can be approximated by some simple functions. Furthermore, the same techniques are applied to determine the tail probability density function for a ratio statistic, and for a sum with more than two lognormally distributed random variables under some stricter conditions. The results yield new insights into the problem of characterization for a sum of lognormally distributed random variables and demonstrate that there is a need to revisit many existing approximation methods.

scholarly journals On The Distribution Of Mixed Sum Of Independent Random Variables One Of Them Associated With Srivastava's Polynomials And H -Function

2014 ◽  
Vol 10 (1) ◽  
pp. 53-62 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Random Variables ◽  
Probability Density Functions ◽  
Independent Random Variables ◽  
Special Cases ◽  
The Laplace Transform ◽  
Srivastava’S Polynomials ◽  
Infinite Range

Abstract In this paper, we obtain the distribution of mixed sum of two independent random variables with different probability density functions. One with probability density function defined in finite range and the other with probability density function defined in infinite range and associated with product of Srivastava's polynomials and H-function. We use the Laplace transform and its inverse to obtain our main result. The result obtained here is quite general in nature and is capable of yielding a large number of corresponding new and known results merely by specializing the parameters involved therein. To illustrate, some special cases of our main result are also given.

scholarly journals Probabilistic Solution of Rational Difference Equations System with Random Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Seifedine Kadry
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Difference Equations ◽  
Random Variables ◽  
Independent Random Variables ◽  
Random Parameters ◽  
Rational Difference Equations ◽  
Probabilistic Solution

We study the periodicity of the solutions of the rational difference equations system of type , (), and then we propose new exact procedure to find the probability density function of the solution, where a, b, and are independent random variables.

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