典型随机动力系统的联合与边缘、条件概率密度的形态分析
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摘要
本文刻画几类典型随机动力系统的二维稳态联合概率密度的形态,并通过引入"横"与"侧"认识角度观察联合概率密度这座"山峰",直观展现其与边缘、条件概率密度三者之间的联系并形成对随机变量独立性的立体认知.从而帮助学生增强对联合概率密度概念的直观认识,加深对三者概率密度相互联系的理解,也有助于相关教师对概率论教学与科研活动互动的促进.
In this paper,the multifarious shapes of steady joint probability densities for several typical stochastic dynamical systems are displayed.From the different horizontal or vertical aspects of observation,the visualized descriptions of joint,marginal and conditional probability densities can be formed in virtue of assimilating joint probability density to a'mountain'.In that case,the viewpoint of an ancient poetry written by Su-Shi,a distinguished writer and poet,is introduced to illustrate the relationship of these three kinds of probability densities.Consequently,the independence of two or more random variables is also explicitly represented.The aforementioned is helpful for enriching the concept of joint probability density and enhancing the interaction of teaching and research related with probability theory.
In this paper,the multifarious shapes of steady joint probability densities for several typical stochastic dynamical systems are displayed.From the different horizontal or vertical aspects of observation,the visualized descriptions of joint,marginal and conditional probability densities can be formed in virtue of assimilating joint probability density to a'mountain'.In that case,the viewpoint of an ancient poetry written by Su-Shi,a distinguished writer and poet,is introduced to illustrate the relationship of these three kinds of probability densities.Consequently,the independence of two or more random variables is also explicitly represented.The aforementioned is helpful for enriching the concept of joint probability density and enhancing the interaction of teaching and research related with probability theory.
引文
[1]徐伟,师义民,秦超英,孙浩.概率论与数理统计[M].北京:高等教育出版社,2008:42-49.
[2]朱位秋,蔡国强.随机动力学引论[M].北京:科学出版社,2017.
[3]R.Yamapi,G.Filatrella,M.A.Aziz-Alaoui,and Hilda A.Cerdeira.Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator[J].Chaos,2012(22):043114.
[4]楼文娟,段志勇,庄庆华.极值风速风向的联合概率密度函数[J].浙江大学学报,2017,51(6):1057-1063.
[5]沙云东,张振,唐金,李纪永.基于联合概率密度的随机声疲劳寿命估算方法[J].振动、测试与诊断,2012,32:32-36.
[6]R.Wang,K.Yasuda.Exact stationary probability density for second order non-linear systems under external white noise excitation[J].Journal of Sound and Vibration,1997,205(5):647-655.
[7]康晓蓉.利用平面域的解析表示求条件概率密度[J].高等数学研究,2018,21(4):85-89.
[8]何典灿.基于多变量联合分布的洪水特性研究[M].华中科技大学,硕士论文,2017.
[9]L.Cai,Y.F.Nie,W.X.Xie,W.W.Zhang.Numerical path integration method based on bubble grids for nonlinear dynamical systems[J].Applied Mathematical Modelling,2013,37(3):1490-1501.
[10]W.X.Xie,W.Xu,L.Cai.Numerical meshfree path integration method for non-linear dynamic systems[J].Applied Mathematics and Computation,2008,197(1):426-434.
[11]G.Muscolino,G.Ricciardi and M.Vasta.Stationary and non-stationary probability density function for non-linear oscillators[J].International Journal of Non-Linear mechanics,1997,32(6):1051-1064.
[12]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2003.
[2]朱位秋,蔡国强.随机动力学引论[M].北京:科学出版社,2017.
[3]R.Yamapi,G.Filatrella,M.A.Aziz-Alaoui,and Hilda A.Cerdeira.Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator[J].Chaos,2012(22):043114.
[4]楼文娟,段志勇,庄庆华.极值风速风向的联合概率密度函数[J].浙江大学学报,2017,51(6):1057-1063.
[5]沙云东,张振,唐金,李纪永.基于联合概率密度的随机声疲劳寿命估算方法[J].振动、测试与诊断,2012,32:32-36.
[6]R.Wang,K.Yasuda.Exact stationary probability density for second order non-linear systems under external white noise excitation[J].Journal of Sound and Vibration,1997,205(5):647-655.
[7]康晓蓉.利用平面域的解析表示求条件概率密度[J].高等数学研究,2018,21(4):85-89.
[8]何典灿.基于多变量联合分布的洪水特性研究[M].华中科技大学,硕士论文,2017.
[9]L.Cai,Y.F.Nie,W.X.Xie,W.W.Zhang.Numerical path integration method based on bubble grids for nonlinear dynamical systems[J].Applied Mathematical Modelling,2013,37(3):1490-1501.
[10]W.X.Xie,W.Xu,L.Cai.Numerical meshfree path integration method for non-linear dynamic systems[J].Applied Mathematics and Computation,2008,197(1):426-434.
[11]G.Muscolino,G.Ricciardi and M.Vasta.Stationary and non-stationary probability density function for non-linear oscillators[J].International Journal of Non-Linear mechanics,1997,32(6):1051-1064.
[12]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2003.