Expected Value of Function of Uncertain Variables

Transcription

1 Journl of Uncertin Systems Vol.4, No.3, pp.8-86, 2 Online t: Expected Vlue of Function of Uncertin Vribles Yuhn Liu, Minghu H College of Mthemtics nd Computer Sciences, Hebei University, Hebei 72, Chin Received 25 July 29; Revised 3 September 29 Abstrct Uncertinty theory is brnch of mthemtics bsed on normlity, monotonicity, self-dulity, countble subdditivity, nd product mesure xioms. Different from rndomness nd fuzziness, uncertinty theory provides new mthemticl model for uncertin phenomen. A key concept to describe uncertin quntity is uncertin vrible, nd expected vlue opertor provides n verge vlue of uncertin vrible in the sense of uncertin mesure. This pper will prove tht the expected vlue of monotone function of uncertin vrible is just Lebesgue-Stieltjes integrl of the function with respect to its uncertinty distribution, nd give some useful expressions of expected vlue of function of uncertin vribles. c 2 World Acdemic Press, UK. All rights reserved. Keywords: uncertin vrible, uncertin mesure, expected vlue, uncertinty distribution Introduction When uncertinty behves neither rndomness nor fuzziness, we cnnot del with this type of uncertinty by probbility theory or fuzzy set theory. In order to del with uncertinty in humn systems, Liu [5] founded n uncertinty theory in 27 tht is brnch of mthemtics bsed on normlity, monotonicity, self-dulity,countble subdditivity, nd product mesure xioms. In order to describe uncertin vrible, Liu [] suggested the concept of first identifiction function nd Liu [] proposed the concept of second identifiction function. Go [2] gve some mthemticl properties of uncertin mesure. You [4] proved some convergence theorems of uncertin sequence. The uncertinty theory hs become new tool to describe subjective uncertinty nd hs wide ppliction both in theory nd engineering. For the detiled expositions, the interested reder my consult the book []. As n ppliction of uncertinty theory, Liu [8] presented uncertin progrmming which is type of mthemticl progrmming involving uncertin vribles, nd pplied uncertin progrmming to system relibility design, fcility loction problem, vehicle routing problem, project scheduling problem, finnce, control nd soon. In ddition, uncertin process ws defined by Liu [6] s sequence of uncertin vribles indexed by time or spce. Furthermore, Liu [7] proposed uncertin clculus tht is brnch of mthemtics for modelling uncertin processes through integrl or differentil equtions involving uncertin vribles. Li nd Liu [3] proposed uncertin logic nd defined the truth vlue s the uncertin mesure tht the uncertin proposition is true. After tht, uncertin entilment ws developed by Liu [9] s methodology for clculting the truth vlue of n uncertin formul vi the mximum uncertinty principle when the truth vlues of other uncertin formuls re given. Furthermore, Liu [7] proposed uncertin inference tht is process of deriving consequences from uncertin knowledge or evidence vi the tool of conditionl uncertinty. Expected vlue opertor for uncertin vribles hs become n importnt role in both theory nd prctice. This pper will discuss the expected vlue of function of uncertin vribles, nd prove tht the expected vlue of monotone function of uncertin vrible is just Lebesgue-Stieltjes integrl of the function with respect to its uncertinty distribution. This pper lso gives some useful expressions of expected vlue of function of uncertin vribles vi the inverse uncertinty distributions. Corresponding uthor. Emil: liu8727@sin.com (Y. Liu).

2 82 Y. Liu nd M. H: Expected Vlue of Function of Uncertin Vribles 2 Preliminry Different from probbility mesure, cpcity [], fuzzy mesure [2], possibility mesure [5] nd credibility mesure [4], Liu [5] proposed concept of uncertin mesure s follows. Definition ([5]) Let Γ be nonempty set, nd let L be σ-lgebr over Γ. Ech element Λ L is clled n event. A set function M : L [, ] is clled n uncertin mesure if (i) M{Γ} = ; (ii) M{Λ } M{Λ 2 } whenever Λ Λ 2 ; (iii) M{Λ} + M{Λ c } = ; (iv) M{ i= Λ i} Σ i= M{Λ i}. Definition 2 ([5]) An uncertin vrible is mesurble function from n uncertinty spce (Γ, L, M) to the set of rel numbers, i.e., for ny Borel set B of rel numbers, the set is n event. {ξ B} = {γ Γ ξ(γ) B} () The uncertinty distribution Φ(x) : R [, ] of n uncertin vrible ξ is defined by Φ(x) = M{γ Γ ξ(γ) x}. (2) An uncertinty distribution Φ is sid to be regulr if its inverse function Φ (α) exists nd is unique for ech α (, ). Definition 3 ([5]) Let ξ be n uncertin vrible. Then the expected vlue of ξ is defined by provided tht t lest one of the two integrls is finite. M{ξ r}dr M{ξ r}dr (3) Theorem ([5]) Let ξ be n uncertin vrible with uncertinty distribution Φ. If the expected vlue E[ξ] exists, then xdφ(x). (4) Theorem 2 ([]) Let ξ be n uncertin vrible with regulr uncertinty distribution Φ. If the expected vlue E[ξ] exists, then 3 Function of Single Uncertin Vrible Φ (α)dα. (5) It is well known tht the expected vlue of function of rndom vrible is just the Lebesgue-Stieltjes integrl of the function with respect to its probbility distribution. For fuzzy vribles, Zhu nd Ji [6] nd Xue et. l. [3] proved the nlogous result when the function is monotone. This pper will show tht the expected vlue of monotone function of uncertin vrible is just the Lebesgue-Stieltjes integrl of the function with respect to its uncertinty distribution. Theorem 3 Let ξ be n uncertin vrible with uncertinty distribution Φ. If f(x) is monotone function such tht the the expected vlue E[f(ξ)] exists, then E[f(ξ)] = f(x)dφ(x). (6) Proof: We first suppose tht f(x) is monotone incresing function. Since the expected vlue E[f(ξ)] is finite, we immeditely hve lim M{ξ y}f(y) = lim ( Φ(y))f(y) =, (7) y + y +

3 Journl of Uncertin Systems, Vol.4, No.3, pp.8-86, 2 83 lim M{ξ y}f(y) = lim Φ(y)f(y) =. (8) y y Assume tht nd b re two rel numbers such tht < < b. The integrtion by prts produces M{f(ξ) r}dr = Using (7) nd letting b +, we obtin In ddition, M{f(ξ) r}dr = M{ξ f (r)}dr = = M{ξ f (b)}f(f (b)) = M{ξ f (b)}f(f (b)) + M{f(ξ) r}dr = Using (8) nd letting, we obtin It follows from (9) nd () tht E[f(ξ)] = f () M{ξ f (r)}dr = f (b) f () f (b) f () f (b) f () M{ξ y}df(y) f(y)dm{ξ y} (9) f () = M{ξ f ()}f(f ()) = M{ξ f ()}f(f ()) M{f(ξ) r}dr = f () f () f () M{ξ y}df(y) f () f () f () M{f(ξ) r}dr M{f(ξ) r}dr = f(y)dm{ξ y} () If f(x) is monotone decresing function, then f(x) is monotone incresing function. Hence The theorem is verified. E[f(ξ)] = E[ f(ξ)] = f(x)dφ(x) = Exmple : Let ξ be positive liner uncertin vrible L(, b). Then its uncertinty distribution is Φ(x) = (x )/(b ) on [, b]. Thus E[ξ 2 ] = x 2 dφ(x) = 2 + b 2 + b. 3 Exmple 2: Let ξ be positive liner uncertin vrible L(, b). Then E[exp(ξ)] = exp(x)dφ(x) = exp(b) exp(). b

4 84 Y. Liu nd M. H: Expected Vlue of Function of Uncertin Vribles Theorem 4 Assume ξ is n uncertin vrible with regulr uncertinty distribution Φ. If f(x) is strictly monotone function such tht the expected vlue E[f(ξ)] exists, then E[f(ξ)] = f(φ (α))dα. () Proof: Suppose tht f is strictly incresing function. It follows tht the uncertinty distribution of f(ξ) is described by Ψ (α) = f(φ (α)). By using Theorem 2, the eqution() is proved. When f is strictly decresing function, it follows tht the uncertinty distribution of f(ξ) is described by Ψ (α) = f(φ ( α)). By using Theorem 2 nd the chnge of vrible of integrl, we obtin the eqution (). The theorem is verified. Exmple 3: Let ξ be nonnegtive uncertin vrible with regulr uncertinty distribution Φ. Then E[ ξ] = Φ (α)dα. (2) Exmple 4: Let ξ be positive uncertin vrible with regulr uncertinty distribution Φ. Then E [ ] = ξ Φ ( α) dα = 4 Function of Multiple Uncertin Vribles Φ dα. (3) (α) Now we ssume tht ξ, ξ 2,, ξ n re uncertin vribles nd f is mesurble function. expected vlue of f(ξ, ξ 2,, ξ n )? In order to nswer this question, let us introduce lemm. Wht is the Lemm ([]) Let ξ, ξ 2,, ξ n be independent uncertin vribles with regulr uncertinty distributions Φ, Φ 2,, Φ n, respectively. If f : R n R is strictly incresing function, then is n uncertin vrible with uncertinty distribution whose inverse function is Ψ(x) = Ψ (α) = f(φ ξ = f(ξ, ξ 2,, ξ n ) (4) sup f(x,x 2,,x n )=x min Φ i(x i ), x R (5) i n (α), Φ 2 (α),, Φ n (α)), < α <. (6) Theorem 5 Assume ξ, ξ 2,, ξ n re independent uncertin vribles with regulr uncertinty distributions Φ, Φ 2,, Φ n, respectively. If f : R n R is strictly monotone function, then the uncertin vrible ξ = f(ξ, ξ 2,, ξ n ) hs n expected vlue provided tht the expected vlue E[ξ] exists. f(φ (α), Φ 2 (α),, Φ n (α))dα (7)

5 Journl of Uncertin Systems, Vol.4, No.3, pp.8-86, 2 85 Proof: Suppose tht f is strictly incresing function. distribution of ξ is described by It follows from Lemm tht the uncertinty Ψ (α) = f(φ (α), Φ 2 (α),, Φ n (α)). Since the expected vlue E[ξ] exists, it follows from Theorem 2 tht Ψ (α)dα = f(φ (α), Φ 2 (α),, Φ n (α))dα which is just (7). When f is strictly decresing function, the uncertinty distribution of ξ is described by Ψ (α) = f(φ ( α), Φ 2 ( α),, Φ n ( α)). Since the expected vlue E[ξ] exists, it follows from Theorem 2 nd the chnge of vrible of integrl tht (7) holds. The theorem is proved. Exmple 5: Let ξ nd η be independent nd nonnegtive uncertin vribles with regulr uncertinty distributions Φ nd Ψ, respectively. Then E[ξη] = Φ (α)ψ (α)dα. (8) Lemm 2 ([]) Let ξ, ξ 2,, ξ n be independent uncertin vribles with regulr uncertinty distributions Φ, Φ 2,, Φ n, respectively. If the function f(x, x 2,, x n ) is strictly incresing with respect to x, x 2,, x m nd strictly decresing with x m+, x m+2,, x n, then ξ = f(ξ, ξ 2,, ξ n ) is n uncertin vrible with uncertinty distribution ( ) Ψ(x) = sup min Φ i(x i ) min ( Φ i(x i )), x R f(x,x 2,,x n)=x i m m+ i n whose inverse function is Ψ (α) = f(φ (α),, Φ m (α), Φ m+ ( α),, Φ n ( α)), < α <. Theorem 6 Assume ξ, ξ 2,, ξ n re independent uncertin vribles with regulr uncertinty distributions Φ, Φ 2,, Φ n, respectively. If the function f(x, x 2,, x n ) is strictly incresing with respect to x, x 2,, x m nd strictly decresing with x m+, x m+2,, x n, then ξ = f(ξ, ξ 2,, ξ n ) hs n expected vlue, provided tht the expected vlue E[ξ] exists. f(φ (α),, Φ m (α), Φ m+ ( α),, Φ n ( α))dα (9) Proof: Since the function f(x, x 2,, x n ) is strictly incresing with respect to x, x 2,, x m nd strictly decresing with x m+, x m+2,, x n, it follows from Lemm 2 tht the uncertinty distribution of ξ is described by Ψ (α) = f(φ (α),, Φ m (α), Φ m+ ( α),, Φ n ( α)). By the existence of expected vlue E[ξ] nd Theorem 2, we get The theorem is proved. Ψ (α)dα = f(φ (α),, Φ m (α), Φ m+ ( α),, Φ n ( α))dα. Exmple 6: Let ξ nd η be two independent positive uncertin vribles with regulr uncertinty distributions Φ nd Ψ, respectively. It follows from Theorem 6 tht [ ] ξ Φ (α) E = η Ψ ( α) dα.

6 86 Y. Liu nd M. H: Expected Vlue of Function of Uncertin Vribles 5 Conclusion This pper proved tht the expected vlue of monotone function of single uncertin vrible is just Lebesgue- Stieltjes integrl of the function with respect to its uncertinty distribution. This pper lso gve some useful expressions of expected vlue of function of multiple uncertin vribles vi inverse uncertinty distributions. References [] Choquet, G., Theory of cpcities, Annls de I institute Fourier, vol.5, pp.3 295, 954. [2] Go, X., Some proprties of continuous uncertin mesure, Interntionl Journl of Uncertinty, Fuzziness Knowledge-Bsed Systems, vol.7, no.3, pp , 29. [3] Li, X., nd B. Liu, Hybrid logic nd uncertin logic, Journl of Uncertin Systems, vol.3, no.2, pp.83 94, 29. [4] Liu, B., nd Y.K. Liu, Expected vlue of fuzzy vrible nd fuzzy expected vlue models,ieee Trnsctions on Fuzzy Systems, vol., no.4, pp , 22. [5] Liu, B., Uncertinty Theory, 2nd Edition, Springer-Verlg, Berlin, 27. [6] Liu, B., Fuzzy process, hybrid process nd uncertin process, Journl of Uncertin Systems, vol.2, no., pp.3 6, 28. [7] Liu, B., Some reserch problems in uncertinty theory, Journl of Uncertin Systems, vol.3, no., pp.3, 29. [8] Liu, B, Theory nd Prctice of Uncertin Progrmming, 2nd Edition, Springer-Verlg, Berlin, 29. [9] Liu, B., Uncertin entilment nd modus ponens in the frmework of uncertin logic, Journl of Uncertin Systems, vol.3, no.4, pp , 29. [] Liu, B., Uncertinty Theory: A Brnch of Mthemtics for Modeling Humn Uncertinty, Springer-Verlg, Berlin, 2. [] Liu, Y., How to generte uncertin mesures, Proceedings of Tenth Ntionl Youth Conference on Informtion nd Mngement Sciences, pp.23 26, 28. [2] Sugeno, M., Theory of Fuzzy Integrls nd its Applictions, Ph.D. Disserttion, Tokyo Institute of Technology, 974. [3] Xue, F., W.S. Tng, nd R.Q. Zho, The expected vlue of function of fuzzy vrible with continuons membership function, Computers nd Mthemtics with Applictions, vol.55, pp , 28. [4] You, C., Some convergence theorems of uncertin sequences, Mthemticl nd Computer Modelling, vol.49, nos.3-4, pp , 29. [5] Zdeh, L.A., Fuzzy sets s bsis for theory of possibility, Fuzzy Sets nd Systems, vol., pp.3 28, 978. [6] Zhu, Y., nd X. Ji, Expected vlue of function of fuzzy vribles, Journl of Intelligent & Fuzzy Systems, vol.7, no.5, pp , 26.