Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation–Water Model in Arid Ecosystems

Zhou, Baoquan; Han, Bingtao; Jiang, Daqing; Hayat, Tasawar; Alsaedi, Ahmed

doi:10.1007/s00332-022-09789-7

Cited by 17 publications

(5 citation statements)

References 42 publications

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“…and let its probability density function [17,19,20] p : I m → R ++ be continuous. Then the probability distribution function [17,23] of the random variable µ is…”

Section: Applications In Probability Theorymentioning

confidence: 99%

“…where χ(T ) = P (R) is the probability [22,[24][25][26][27] of the random event (20) hold, then we have…”

Section: Applications In Probability Theorymentioning

confidence: 99%

“…The probability density function [17,19,20] of random variable is a basic concept in probability and statistics theories. In [17], the authors studied the properties of the probability density function which is of the logarithmic concavity and displayed their applications in the analysis of variance and the higher education.…”

Section: Introductionmentioning

confidence: 99%

“…The research tools of the paper include the functional analysis, discrete mathematics, probability [17,19,20,[22][23][24][25][26][27], convex geometry [28], linear algebra [2], inequality and the mean value [15,16,18,27,[29][30][31][32][33][34] theories, especially the mean value theory. The research methods of the paper are based on the mathematical induction [2,17,18,32], reorder method [2] and the dimension reduction method [32].…”

Section: Introductionmentioning

confidence: 99%

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Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Wen

Han

Yuan

2023

Preprint

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This paper will establish Chebyshev-Jensen-type inequalities which involving the Chebyshev products as follows:\[\langle X^{1}, X^{2}, \ldots, X^{m}\rangle_{\chi}\geq \left[ X^{1}, X^{2}, \ldots, X^{m}\right]_{\chi}\geq \chi\left( \overline{X^{1}}, \overline{X^{2}}, \ldots, \overline{X^{m}}\right)\]and\[\langle f_{1}, f_{2}, \ldots, f_{m}\rangle_{\chi}\geq \left[f_{1}, f_{2}, \ldots,f_{m}\right]_{\chi}\geq \chi\left( \overline{f_{1}}, \overline{f_{2}}, \ldots, \overline{f_{m}}\right),\]as well as display their applications in probability theory and surround system, especially, the observation angles inequalities are obtained as follows:\[1\geq \frac{1}{n}\sum_{i=1}^{n}P\left(R_{i}\right)\geq \frac{1}{n^{m}}\sum_{1\leq i_{1},\ldots,i_{m}\leq n}P(R_{i_{1},\ldots,i_{m}})\geq\left(\frac{2}{n}\right)^{m+|\alpha|}.\]The research methods of the paper are based on the mathematical induction, reorder method and the dimension reduction method. 2010 MSC: 26D15, 26E60, 51K05, 52A40

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“…and let its probability density function [17,19,20] p : I m → R ++ be continuous. Then the probability distribution function [17,23] of the random variable µ is…”

Section: Applications In Probability Theorymentioning

confidence: 99%

“…where χ(T ) = P (R) is the probability [22,[24][25][26][27] of the random event (20) hold, then we have…”

Section: Applications In Probability Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Wen

Han

Yuan

2023

Preprint

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“…Numerical methods for solving SDEs [2][3][4][5][6][7] make it possible to find approximate solutions and also to estimate their probability densities. As such an estimate, the histogram is usually considered [11][12][13][14][15], which is a statistical analogue for the probability density of the SDE solution. Note that histograms are used not only when solving SDEs but also when analyzing various data [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Conditional Optimization of Algorithms for Estimating Distributions of Solutions to Stochastic Differential Equations

Averina

2024

Mathematics

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This article discusses an alternative method for estimating marginal probability densities of the solution to stochastic differential equations (SDEs). Two algorithms for calculating the numerical–statistical projection estimate for distributions of solutions to SDEs using Legendre polynomials are proposed. The root-mean-square error of this estimate is studied as a function of the projection expansion length, while the step of a numerical method for solving SDE and the sample size for expansion coefficients are fixed. The proposed technique is successfully verified on three one-dimensional SDEs that have stationary solutions with given one-dimensional distributions and exponential correlation functions. A comparative analysis of the proposed method for calculating the numerical–statistical projection estimate and the method for constructing the histogram is carried out.

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Dynamics and density function of a stochastic COVID-19 epidemic model with Ornstein–Uhlenbeck process

Shi

Jiang

2023

Nonlinear Dyn

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Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation–Water Model in Arid Ecosystems

Cited by 17 publications

References 42 publications

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Conditional Optimization of Algorithms for Estimating Distributions of Solutions to Stochastic Differential Equations

Dynamics and density function of a stochastic COVID-19 epidemic model with Ornstein–Uhlenbeck process

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