DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES

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1 MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS Vol, No, 4, pp5- DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES Yogchao Hou* ad Weicai Peg Departet of Matheatical Scieces, Chaohu Uiversity, Ahui 38, Chia Eail: ABSTRACT I order to deal with ideteriacy data ivolvig ucertaity ad radoess, ucertai rado variable is ivestigated by ay scholars As a extesio of distace betwee ucertai variables, the defiitio of distace betwee ucertai rado variables is proposed i this paper The, soe forulas are provided to calculate distaces betwee particular types of ucertai rado variables Keywords: Ucertaity theory, Ucertai rado variable, Chace easure, Distace INTRODUCTION I order to deal with ideteriacy pheoeo i daily life, several axioatic systes have bee fouded I 933, Kologorov [5] fouded a axioatic syste of probability theory to describe rado pheoea If there are eough historical data, we ca eploy probability theory to estiate probability distributio Soeties, it is difficult to collect observed data whe soe uexpected evets occur I this case, people have to ivite experts to estiate the belief degree of each evet s occurrece However, soe couterituitive cosequeces ay occur if we eploy probability theory or fuzzy set theory to odel the belief degree [] For dealig with belief degree legitiately, a axioatic syste aed ucertaity theory was proposed by Liu [7] i 7 I additio, the product ucertai easure was defied by Liu [9] i 9 The cocept of ucertai variable ad ucertaity distributio were proposed by Liu [7] The, a sufficiet ad ecessary coditio of ucertaity distributio was proved by Peg ad Iaura [8] i To describe the relatioship betwee ucertai easure ad ucertai distributio, a easure iversio theore was preseted by Liu [] fro which the ucertai easures of soe evets would be calculated via the ucertaity distributio After proposig the cocept of idepedece [9], Liu [] preseted the operatioal law of ucertai variables The cocepts of expected value, variace, oets ad distace of ucertai variable were proposed by Liu [] Besides, a useful forula was preseted by Liu ad Ha [3] to calculate the expected values of ootoe fuctios of ucertai variables I order to characterize the ucertaity of ucertai variables, Liu [9] proposed the cocept of etropy i 9 After that, Dai ad Che citedai proved the positive liearity of etropy ad gave soe forulas to calculate the etropy of ootoe fuctio of ucertai variables Che ad Dai [] discussed the ethod to select the ucertaity distributio usig the axiu etropy priciple I order to ake a extesio of etropy, Che, Kar ad Ralescu [] proposed a cocept of cross-etropy for coparig a ucertaity distributio agaist a referece ucertaity distributio I 3, Liu [4] proposed chace theory by defiig ucertai rado variable ad chace easure i order to describe a syste that ivolved both ucertaity ad radoess Soe related cocepts of ucertai rado variables such as chace distributio, expected value, ad variace were also preseted by Liu [4] As a iportat cotributio to chace theory, Liu [5] proposed a basic operatioal ethod of ucertai rado variables After that, ucertai rado variables were discussed widely A law of large ubers was preseted by Yao ad Gao [3] Besides, Yao ad Gao [] proposed a ucertai rado process As applicatios of chace theory, Liu [5] proposed ucertai rado prograig Ucertai rado risk aalysis was preseted by Liu [6] Besides, chace theory was applied ito ay fields ad ay achieveets were obtaied, such as ucertai rado reliability aalysis (We ad Kag []), ucertai rado logic (Liu eliuy3d), ucertai rado graph (Liu []), ad ucertai rado etwork (Liu []) I this paper, the distace betwee ucertai rado variables is studied I the first sectio, the ucertaity theory ad ucertai rado variables are itroduced I the followig sectio, the defiitio of distace betwee ucertai rado variables is preseted ad soe forulas are proposed to calculate the distace betwee specific type s ucertai rado variables I additio, the ethod is illustrated by exaples PRELIMINARY As a brach of axioatic atheatics, ucertaity theory ais to deal with hua ucertaity We will first preset soe basic cocepts of ucertai theory ad chace theory 5

2 Ucertai variables Suppose that is a oepty set ad L is a -algebra o Each eleet i L is called a evet A set fuctio M fro L to [ ] that satisfies orality axio, duality axio, subadditivity axio (Liu [7]) ad product axio (Liu [9]) is called a ucertai easure I geeral, ( L M) is called a ucertaity space A ucertai variable is defied as a easurable fuctio fro ( L M) to R That is to say, the set ( B) { ( ) B} is a evet, for ay Borel set B of real ubers Liu [7] preseted the defiitio of ucertaity distributio to represet ucertai variables, i which is defied as ( x) M{ x} for ay real uber x Defiitio (Liu [7]) If the ucertaity distributio of a ucertai variable is if xa ( x) ( x a) ( b a) if a x b if x b Where a ad b are real ubers with a b, the is called liear ucertai variable ad its ucertaity distributio is deoted by L( a b) Defiitio (Liu [9]) Let variables If M ( ) M{ } i Bi i Bi i i For ay Boral sets say be ucertai B B B of real ubers, the we are idepedet ucertai variables f ( x x x ) be strictly Theore (Liu [7]) Let ootoe icreasig with respect to x x x ad strictly x x x Let ootoe decreasig with respect to be idepedet regular ucertai variables ad f ( ) The, the iverse ucertaity distributio of ( ) ca be calculated by ( ) f ( ( ) ( ) ( ) ( )) I which ( ) ( ) ( ) are iverse, respectively ucertaity distributios of I order to describe the ea value of a ucertai variable by ucertai easure, Liu [7] defied the expected value of as () E[ ] M{ r} M{ r} Provided that at least oe of the two itegrals is fiite I additio, the expected value ca be calculated by () E[ ] ( ( x)) ( x) Furtherore, the expected value of a fuctio of ucertai variables ca be calculated by iverse ucertaity distributios Theore (Liu ad Ha [3]) Let be idepedet regular ucertai variables If the coditios of Theore hold, the the expected value of ucertai variable f ( ) is (3) E[ ] f ( ( ) ( ) ( ) ( )) d Defiitio 3 (Liu [7]) Let ad be two ucertai variables The distace betwee ad is defied as d( ) E[ ] M{ r} (4) Besides, the distace d( ) satisfies o egativity, idetificatio, syetry ad triagular iequality Ucertai rado variables I 3, Liu [4] first proposed chace theory, which is a atheatical ethodology for odelig coplex systes with both ucertaity ad radoess, icludig chace easure, ucertai rado variable, chace distributio, operatioal law, expected value ad so o The chace space is refer to the product ( L M) ( A Pr), i which ( L M) is a ucertai space ad ( A Pr) is a probability space Defiitio 4 (Liu [4]) Let ( L M) ( A Pr) be a chace space, ad let L A be a evet The the chace easure of is defied as Ch{ } Pr{ M{ ( ) } r} Liu ([4]) proved that a chace easure satisfies orality, duality, ad ootoicity properties, that is (a) Ch{ }, Ch{ } ; c (b) Ch{ } Ch{ } For ay evet ; Ch{ } Ch{ } For ay evet (c) Lea (Hou ([4])) The chace easure is sub additive That is, for ay coutable sequece of evets have Ch i i i Ch i, we Roughly speakig, a ucertai rado variable is a easurable fuctio of ucertai variables ad rado 6

3 variables Defiitio 5 (Liu [4]) A ucertai rado variable is a fuctio fro a chace space ( L M) ( A Pr) to the set of real ubers ie, {( ) ( ) B} is a evet for ay Boral set B Note that a ucertai rado variable ( ) is a bivariate fuctio o Specifically, both rado variables ad ucertai variables are degeerated ucertai rado variables The ucertai rado arithetic is defied as follows Defiitio 6 (Liu [4]) Let f R R be a easurable be ucertai rado variables o fuctio, ad the chace space ( LM) ( A Pr) The, f ( ) is a ucertai rado variable defied as ( ) f ( ( ) ( ) ( )), For all ( ) ( ) Defiitio 7 (Liu [4]) Let be a ucertai rado variable The chace distributio of is defied by ( x) Ch{ x} for ay x R Theore 3 (Liu [5]) Let be idepedet rado variables with probability distributios, ad let be idepedet ucertai variables with ucertaity distributios Upsilo respectively The the ucertai rado variable f ( ) Has a chace distributio ( x) F( x y y y )d ( y )d ( y ) d ( y ) R F( x y y y ) is the ucertaity distributio f ( y y y ) Where of the ucertai variable Ad is deteried by its iverse fuctio F ( y y y ) f ( y y y ( ) ( ) ( ) ( )) k k Provided that f ( ) is a strictly icreasig fuctio with respect to k decreasig fuctio with respect to ad strictly k k Defiitio8 (Liu [4]) Let be a ucertai rado variable The its expected value is defied by, E[ ] Ch{ r} Ch{ r} Provided that at least oe of the two itegrals is fiite Defiitio9 (Liu [4]) Let be a ucertai rado variable with chace distributio If the expected value of exists, the E[ ] ( ( x)) ( x) be idepedet Theore4 (Liu [4]) Let rado variables with probability distributios, ad let be idepedet ucertai variables with ucertaity distributios Upsilo respectively The the ucertai rado variable f ( ) Has a expected value R k E[ ] f ( x y y ( ) ( ) ( ) ( ))dd ( y ) d ( y ) k Provided that icreasig fuctio with respect to k ad strictly f ( ) is a strictly decreasig fuctio with respect to k k 3 DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES Defiitio The distace betwee ucertai rado variables ad is defied as d( ) E[ ] That is, d( ) Ch{ r} Reark If the ucertai rado variables ad de geerate to ucertai variables, the d( ) E[ ] Ch{ r} M{ r} It eas that the defiitio of distace betwee ucertai rado variables is cosistet with ucertai variables Reark If the ucertai rado variables ad degeerate to rado variables, the d( ) E[ ] Ch{ r} Pr{ r} It eas that the defiitio of distace betwee ucertai rado variables is cosistet with rado variables Theore Let ad 3 be ucertai rado (5) 7

4 variables, ad let d( ) be the distace The we have (a)(no egativity) d( ) ; (b)(idetificatio) d( ) if ad oly if ; (c)(syetry) d( ) d( ) ; (d)(triagle Iequality) d( ) d( 3) d( 3) Proof The proofs of parts (a), (b) ad (c) are trivial Now we prove the part (d) By usig the defiitio of distace ad Lea, we get d( ) E[ ] Ch{ r} Ch{ r} 3 3 Ch{( r ) ( r )} 3 3 Ch{( r)} Ch{( r)} 3 3 E[ ] E[ ] d( ) d( ) Exaple Let { 3} Defie M{ } M{ } ad M{ } for ay subset ( ) Let { } Defie Pr{ } 3 Pr{ } 3 We set ucertai rado variables ad as follows, if if ( ) ( ) 3 if if I which if if ( ) if ( ) if if 3 if 3 It is easy to verify that ad d( 3) Thus 3 d( ) ( d( 3) d( 3)) d( ) 3, d( 3) I the followig, we will discuss the ethod to obtai distace fro chace distributios Let ad be ucertai rado variables with chace distributios ( x) ad ( has a chace distributio ( x), the the distace is d( ) Ch{ x} Ch{( x) ( x)} ( Ch{ x} Ch{ x}) ( ( x) ( x)) We stipulate that the distace betwee ad is (6) d( ) ( ( x) ( x)) Reark 3 Metio that (5) is ot a precise forula but a stipulatio The calculatio forula of distace betwee ucertai rado variables i the rest of this paper is refer to (5) Reark 4 Let ad be rado variables with probability distributios ( x) ad ( x), respectively If has a probability distributio ( x), the the distace betwee ad is d( ) Ch{ x} Pr{ x} Pr{( x) ( x) ( x)} ( Pr{ x} Pr{ x} Pr{ x}) ( ( x) ( x)) That eas (5) is a precise forula whe the ucertai rado variables degeerate to rado variables Theore Let p be idepedet rado variables with probability distributios ( x) ( x) p, ad let be idepedet ucertai variables with ucertaity distributios q ( x) ( x) ( x) q respectively Let ad be two ucertai rado variables, i which f ( ) f ( ) p The, the distace betwee ad is d( ) ( ( x) ( x)) q [ F( x y y )d ( ) d ( ) p R p y p y (7) p F( x y y )d ( y ) d ( y )] p R p p p 8

5 Where F( x y y y p) is the ucertaity distributio of the ucertai variable f ( y y y ) f ( y y y ) p q Ad is deteried by its iverse fuctio F ( y y y ) f y y y ( ) ( ) ( ) ( ) p k k f y y y ( ) ( ) ( ) ( ) p t t q provided that f( ) is a strictly icreasig fuctio with respect to s ad strictly decreasig fuctio with respect to s ad f ( ) is a strictly icreasig fuctio p q with respect to t ad strictly decreasig fuctio with respect to t Proof It follows fro Theore 3 iediately Corollary Let be a rado variable with probability distributio ( x) ad be a ucertai variable with ucertaity distributio ( x) Let ( x) be the chace distributio of, ad the we have d( ) ( ( x) ( x)) ( x y) d( y) ( x y) d( y) Proof Note that ( x) ( x y) d( y) It follows fro Theore iediately Exaple Let be a rado variable with probability distributio ( x) ad c be a real uber with ucertaity distributio ( x) Suppose that ( x) represets the chace distribut io of c Accordig to Corollary, we have ( x) ( x y) d ( y) d ( y) ( c x) I which cx if xc ( x) if x c The we have d( c) ( ( x) ( x)) ( c x) ( c x) Exaple3 Let c be a real uber with probability distributio ( x) ad be a ucertai variable with ucertaity distributio ( x) Suppose that ( x) represets the chace distributio of c Accordig to Corollary, we have ( x) ( x y) d( y) ( x c) if xc ( x) if x c The we have d( c ) ( ( x) ( x)) ( c x) ( c x) Exaple 4 Let b ad c be two real ubers with distributio fuctios ( x) ad ( x) Suppose that ( x) represets the chace distributio of b c Accordig to Corollary, we have if x b c ( x) ( x y) d ( y) ( x c) if x b c I which if x c if x b ( x) ( x) if x c if x b The we have d( cb) ( ( x) ( x)) b c It eas that the defiitio of distace betwee ucertai rado variables is cosistet with real ubers Corollary Let ad be two idepedet rado variables with probability distributios ( x) ad ( x), respectively Let ad be two idepedet ucertai variables with ucertaity distributios ( x) ad ( x) Suppose that ( x) represets the chace distributio ( ) ( ) The we have of d( ) ( ( x) ( x)) R R (8) Where F ( y y ) y y ( ) ( ) is the F( x y y ) d ( y ) d ( y ) F( x y y ) d ( y ) d ( y ) iverse distributio of the ucertai variable y y Proof It follows fro Theore iediately Exaple 5 Let be a rado variable with probability distributio ( x) ad be a ucertai variable with ucertaity distributio ( x), i which exp( x) if x ( x) if x if x ( x) x if x if x We have I which 9

6 x exp( x) if x ( x y) d( y) if x Ad ( x y) d ( y) exp( x) exp( x) x Accordig to Corollary, we have d( ) ( x y) d( y) ( x y) d( y) ( ) ( ) ( ) ( ) x exp x exp x exp x exp x exp( ) exp( ) CONCLUSIONS I this paper, the cocept of distace betwee ucertai variables was expaded to ucertai rado variables Based o the subadditivity of chace easure ad expected value of ucertai rado variables, several properties of the distace were proved The, the effectiveess of this ethod was illustrated by a exaple ACKNOWLEDGMENTS This work was supported by the Natioal Natural Sciece Foudatio of Chia Grat (No67344) ad the Foudatio of Ahui Educatioal Coittee (No KJ4A74, No KJB5) REFERENCES Che X ad Dai W, Maxiu Etropy Priciple for Ucertai Variables, Iteratioal Joural of Fuzzy Systes, vol3, No3, pp3-36, Che X, Kar S ad Ralescu D, Cross-etropy Measure of Ucertai Variables, Iforatio Scieces, Vol, pp53-6, 3 Dai W ad Che X, Etropy of Fuctio of Ucertai Variables, Matheatical ad Coputer Modellig, vol 55, No3-4, pp754-76, 4 Hou Y, Subadditivity of Chace Measure, Joural of Ucertaity Aalysis ad Applicatios, vol:4, 4 5 Kologorov A N, Grudbegriffe der Wahrscheilich- Keitsrechug, Julius Spriger, Berli, Li X ad Liu B, O Distace betwee Fuzzy Variables, Joural of Itelliget ad Fuzzy Systes, vol9, No3, pp97-4, 8 7 Liu B, Ucertaity Theory, d Editio, Spriger-Verlag, Berli, 7 8 Liu B, Theory ad Practice of Ucertai Prograig, d Editio, Spriger-Verlag, Berli, 9 9 Liu B, Soe Research Probles i Ucertaity Theory, Joural of Ucertai Systes, vol3 (), pp3-, 9 Liu B, Ucertaity Theory: A Brach of Matheatics for Modelig Hua Ucertaity, Spriger-Verlag, Berli, Liu B, Why Is There a Need for Ucertaity Theory, Joural of Ucertai Systes, vol6, No, pp 3-, Liu B, Ucertai Rado Graphs ad Ucertai Rado Networks, Techical Report, 3 3 Liu Y ad Ha M, Expected Value of Fuctio of Ucertai Variables, Joural of Ucertai Systes, vol4, No3, pp8-86, 4 Liu Y, Ucertai Rado Variables: A Mixture of Ucertaity ad Radoess, Soft Coputig, vol7, No4, pp65-634, 3 5 Liu Y, Ucertai Rado Prograig with Applicatios, Fuzzy Optiizatio ad Decisio Makig, vol, No, pp53-69, 3 6 Liu Y, Risk Idex i Ucertai Rado Risk Aalysis, 7 Liu Y, Ucertai Rado Logic ad Ucertai Rado Etailet, Techical Report, 3 8 Peg Z ad Iwaura K, A Sufficiet ad Necessary Coditio of Ucertaity Distributio, Joural of Iterdiscipliary Matheatics, vol3, No3, pp77-85, 9 Sheg Y, Stability i the P-Th Moet for Ucertai Differetial Equatio, Joural of Itelliget ad Fuzzy Systes, We M ad Kag R, Reliability Aalysis i Ucertai Rado Syste, Yag L ad Liu B, O Cotiuity Theore for Characteristic Fuctio of Fuzzy Variable, Joural of Itelliget ad Fuzzy Systes, vol7, No3, pp 35-33, 6 Yao K ad Gao J, Law Of Large Nubers for Ucertai Rado Variables, 3 Yao K ad Gao J, Soe Cocepts ad Theores of Ucertai Rado Process, 4 Yao K ad Che X, A Nuerical Method for Solvig Ucertai Differetial Equatios, Joural of Itelliget ad Fuzzy Systes, 5 Zhu Y ad Ji X, Expected Values of Fuctios of Fuzzy Variables, Joural of Itelliget ad Fuzzy Systes, vol7, No5, pp47-478, 6