Promote the Use of Two-dimensional Continuous Random Variables Conditional Distribution

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1 Promote te Use of Two-dimensional Continuous Random Variables Conditional Distribution Feiue Huang Deartment of Economics Dalian University of Tecnology Dalian 604 Cina Tel: Ceng Li Deartment of Economics Dalian University of Tecnology Dalian 604 Cina Abstract Conditional distribution reflects te deendency lin among random variables but two-dimensional random variables Conditional Distribution as some limitations. In order to ric te content of conditional distribution tis aer gives te etension of conditional distribution and eamles in te case of continuous random variables. For te given definition of conditional distribution of tree-dimensional continuous random variables it also gives te roof. Tis article obtains te etension strictly in accordance wit te definition of two-dimensional random variables and it uses te teory of conditional robability to get te roof. So it can get conditional distributions after canging te condition to enric te contents of te conditional distribution. Keywords: Promote te use Continuous random variables Conditional distribution Multi-dimensional random variables Introduction Y are mainly divided into two tyes: indeendence and deendence. Te more evaluations and Y ave te more conditional distributions will be tere. Eac conditional distribution describes one state-secific distribution from a side. So te contents of conditional distribution are ricer and its alications are broader. For in many issues values of concerned random variables tend to influence eac oter tis maes conditional distribution as a owerful tool for studying deendencies among variables(mao Si-song Ceng Yi-ming Pu iao-long. 004). Conditional distribution for random variables derives from conditional robability for random events so tere is a close relationsi between te two and te aroaces to andle tem are te same but conditional distribution is more comle to deal wit(feng Tai Liu De-yin. 985)( Liang Yi. 998). Te relationsis of two-dimensional random variables ( ) In recent years te main researc direction in Cina is te researc of conditional eigenvalues and te etension and alication of conditional distribution. Conditional eigenvalues are mainly ointed to conditional eectation wic is eectation under conditional distribution. In tis area te main contributions are Wang Ceng ou Hailei gave te definition and researced te caracteristics of random variables conditional articular function based on measurement and integral teory(wang Ceng ou Hai-lei. 008); ang Mei used minimum mean-squared error to solve one ind roblems of best rediction and too eamles to analye te alication of conditional eectation in ractical rediction roblems(ang Mei. 006); u ui and Wu Guogeng educed total robability formula of discrete and continuous random variables based on conditional eectation and indicative function I A of random events A(u Hui Wu Guo-gen. 004). For etension of conditional distribution rimacy researces are Ceng Feiyue generalied te condition distribution of te oisson rocess arrival time(cen Fei-yue. 006); Yang Jinging etc. investigated te marginal recursive equations on ecess-of-loss reinsurance treaty under te assumtion tat te number of claims belongs to te family consisting of Poisson binomial and negative binomial and tat te severity distribution and bounded continuous density function(yang Jing-ing Wang iao-qian Ceng Si-ong. 006); Hu Duaning gave a eress formula of distribution for ellitically contoured matri distributions and sowed tat condition distribution of ellitically contoured matri distribution is ellitically contoured distribution yet(hu Duan-ing. 00). In te above study tey focused on te alication of te conditional distribution sowing te researc significance of te conditional distribution but tere is few studies researcing te nature of te conditional distribution. In comarison te studies of conditional distribution abroad are more in-det. For eamle Rodney C.L. Wolff etc studied te metods of evaluating conditional distribution function; Jusan Bai investigated te dynamic model of testing arametric conditional distributions; Peter Hall and Qiwei Yao discussed aroimating 50 ISSN E-ISSN

2 conditional distribution function using dimension reduction; Bruce E. Hansen studied nonarametric estimation of smoot conditional distributions; Persi Diaconis Bernd Sturmfels analyed conditional distributions using algebraic algoritms for samling. Tey ave sown tat te researces of conditional distribution are multi-faceted and more comle wile mae against undergraduate teacing. In tis resect te aer begins to discuss and analye from te basic content of conditional distribution and educes general formulas wit certain conditions on te basis of te definition of conditional distribution. It first starts form te two-dimensional random variable conditional distribution in te case of continuous random variables and canges te given conditions to obtain te etensions of conditional distribution and ten gives etensions of conditional distribution wen tere are tree-dimensional random variables. And it roves te definition of te given tree-dimensional continuous random variables conditional distribution. Tis aer is to solve te conditional distribution of multidimensional random variables under te given conditions and its results can be used for teacing eending te nowledge of te conditional distribution and facilitating eole s calculations.. Etension of continuous random variables conditional distribution Y and let Etension. Set Y ( y ) te joint density function of two-dimensional random variables + Y of wic te density function is ( ). ( y ) + Y te conditional density function of is Y ( ) ( ) function of Y is Y ( y y) Y( y ). u To rove: From te Variable Transformation note u its inverse function is v + y y v u y u u J. y 0 v v So te joint density function of ( ) UV is UV ( ) Y ( ( ) ( )) is ( ) Y( ) ( + ) lim ( + ) P( + ) lim 0 P( + ) are continuous everywere. Wen and te conditional density ten uv uv y uv J. () Tat te joint density function of ( ) P Y P lim + + ( ) d dd. + ( ) d d lim. () + d From mean value teorem of integrals we get lim + d () + lim ( ) d ( ). (4) ( ) Y( ) So P ( + Y ) d d were te conditional density Publised by Canadian Center of Science and Education 5

3 function is Y ( ) ( ). Eemle. Set two-d random variable ( ) ( 0) + Y u to solve te condition density of and Y. Y N μ μ σ σ and + Y U. Given te condition of To solve: For te marginal distribution of two-dimensional normal distribution is one-dimensional normal Y N μ σ. distribution vi. N ( μ ) σ ( ) Because and Y are indeendent so U Y N( μ μ σ σ) U is U ( u) ( σ) ( u μ μ ) were te density function of e. (5) π σ + σ + σ For and Y are indeendent we get u u Y Y ( μ) ( uμ) σ σ e e πσ πσ ( μ ) uμ σ σ e (6) πσ σ ( ) ( ) u y y u y y Y Y ( uyμ ) ( yμ ) σ σ e e πσ πσ ( uyμ ) yμ σ σ e. (7) πσ σ So ( u) U ( ) ( u) u Y U ( μ) ( uμ) σ σ e πσσ e π σ ( u μ μ ) ( + σ) σ + σ ( ) ( u ) ( u ) π σ + σ μ μ μ μ e + ; (8) πσσ σ σ σ + σ YU ( y u) ( ) ( u) u y y Y U ( σ) ( uyμ) ( yμ) σ σ e πσσ ( u μμ) e π σ + σ + σ ( u y ) ( y ) ( u ) π σ + σ μ μ μ μ e +. (9) πσσ σ σ σ + σ According to te formula it is easy to be drawn. 5 ISSN E-ISSN

4 . Proof of tree-dimensional continuous random variables conditional distribution Definition: Set tree-dimensional continuous random variables ( ) ( ). Its density function f and eac marginal density function are continuous everywere. Given te condition of and te conditional distribution of is ( ) F ( ) ( ) f d f d (0) and conditional density function is f ( ) f ( ) f d (M Fes. 96). () To rove: According to te roof of two-dimensional continuous random variables conditional distribution we can give te conditional distribution of tree-dimensional continuous random variables. ( ) limlim ( + + ) P P 0 0 ( + + ) P( + + ) P lim lim lim lim f ( u v t) dvdudt + + f u v dvdu f ( uvtdv ) du dt ( u v) dv du + + lim lim f. () From mean value teorem of integrals we get + lim f ( uvtdv ) f( u t) 0 () + lim f ( u tdu ) f( t) (4) + lim f uvdv f ( u ) 0 (5) + lim f u du f. (6) Ten we can get conditional distribution function ( t) ( ) f P dt f ( ) (7) ( ) F f ( t) f d f ( ) f d dt. (8) Tere is following conditional density function f ( ) f ( ). (9) f d Publised by Canadian Center of Science and Education 5

5 In a similar way under te condition of conditional distribution and conditional density of and are f dd (0) f ( ) dd f ( ) ( ). () F f ( ) f d d Eamle Tree-dimensional continuous random variables ( ) wic te density function is + y + r ; ( y ) 4π r 0 + y + > r. I) Given te condition of ( ) Y to solve conditional distribution of. Y subject to Uniform distribution on D of To solve: We can get te density function of ( Y ) from te subject: r y dy r y + y r ( ) ( ) r y Y y y d ; 4πr πr 0 oters. So wen + y r tere is ( y ) ( y Y ) r y Y ( y) 0 II) Given te condition of to solve conditional distribution of ( ) () r y r y ; () oters. Y. To solve: We can get te density function of from te subject: π r ( r ) ( r r ) 0 0 y ddy ρd ρ ; 4π r 4r 0 oters. So wen r r tere is ( y ) + y r ; ( y ) π ( r ) Y 0 oters.. Conclusions Tis aer mainly discusses conditional distributions of multidimensional random variables and its related eamles given certain conditions in te case of continuous situation. It canges te original condition of one fied variable into more comle conditions for eamle te condition tat te sum of two variables is fied in two-dimensional situation wic is easy to solve wit discrete random variables and can be etended in continuous situation. Tis aer gives te roof. Wile in te condition of tree-dimensional continuous random variables tis article roves one variable s conditional distribution wen te oter two are fied to mae te conditional distribution teory recise. Tis article gets te above results strictly according to two-dimensional random variables conditional distribution and alies te teory of conditional robability in te roof. Conditional distribution can be alied in te life and wor to resolve ractical roblems. Te alication of using conditional distribution teory to carry out scientific analysis and calculations wit real data is an (4) 5) 54 ISSN E-ISSN

6 imortant reflection of te usefulness of conditional distribution. In addition for tere are more and more discussions about multidimensional random variables in reality tis aer etends te conditional distribution to rovide new train of tougt for te researc in some etent and it aims at enricing te content of conditional distribution deeening te understanding of it and alying it well in ractical. Acnowledgements Tis wor is suorted by te funds roject under te Ministry of Education of te PRC for young eole wo are devoted to te researces of umanities and social sciences under Grant No. 09YJC79005; software + researc of Dalian University of Tecnology under Grant No References Bruce E. Hansen. Nonarametric Estimation of Smoot Conditional Distributions [EB/OL]. (004-05)[ ]. tt:// Cen Fei-yue. (006). Generaliation of te Condition Distribution of te Poisson Process Arrival Time. Journal of Cangsa University of Electric Power (Natural Science) 006 (): Feng Tai Liu De-yin. (985). Conditional Probability and Conditional Distribution. Distance Education in Cina 985 (0):8-. Hu Duan-ing. (00). Te Condition Distribution of Ellitically Contoured Matri Distributions. Communication on Alied Matematics and Comutation 00 5():4-44. Jusan Bai. (00). Testing Parametric Conditional Distributions of Dynamic Models. Te Review of Economics and Statistics 00 85(): Liang Yi. (998). Conditional Probability and Conditional Distribution. Journal of Higer Corresondence Education (Natural Sciences) 998(5):9-5. M Fes. (96). Probability and Matematical Statistics. Wang fu-bao translate. Sangai: Science & Tecnology Press of Sangai 96. Mao Si-song Ceng Yi-ming Pu iao-long. (004). Probability Teory and Matematical Statistics Tutorial. Beijing: Advanced Education Press 004. Persi Diaconis Bernd Sturmfels. (998). Algebraic algoritms for samling from conditional distributions. Te Annals of statistics 9986():6-97. Peter Hall Qiwei Yao. (005). Aroimating Conditional Distribution Functions Using Dimension Reduction. Te Annals of statistics 005 (): Rodney C.L. Wolff Qiwei Yao Peter Hall. (999). Metods for Estimating a Conditional Distribution Function. Journal of te American Statistical Association (445):54-6. Wang Ceng ou Hai-lei. (008). Some Proerties of Conditional Caracteristic function. Journal of Cina Jiliang University 008 9(4):60-6. u Hui Wu Guo-gen. (004). Full Probability Formula Based on te Conditional Eectation and its Allications]. Journal of East Cina Institute of Tecnology 004 7():9-95. Yang Jing-ing Wang iao-qian Ceng Si-ong. (006). Conditional Recursive Equations on Ecess-of-Loss Reinsurance. Alied Matematics and Mecanics 006 7(8):9-99. ang Mei. (006). Using Conditional Eectation to solve Problems of Best Prediction. Journal of Sanani Institute of Education 006 ():8-84. Publised by Canadian Center of Science and Education 55