Uncertain random variables: a mixture of ... - Semantic Scholar
Soft Comput (2013) 17:625–634 DOI 10.1007/s00500-012-0935-0
FOCUS
Uncertain random variables: a mixture of uncertainty and randomness Yuhan Liu
Published online: 27 September 2012 Springer-Verlag 2012
Abstract In many cases, human uncertainty and objective randomness simultaneously appear in a system. In order to describe this phenomena, this paper presents a new concept of uncertain random variable. To measure uncertain random events, this paper also combines probability measure and uncertain measure into a chance measure. Based on the tool of chance measure, the concepts of chance distribution, expected value and variance of uncertain random variable are proposed. Keywords Uncertainty theory Probability theory Uncertain random variable Chance measure
1 Introduction Probability theory (Kolmogorov 1933) is a branch of mathematics for studying the behavior of random phenomena. Although probability theory has been applied widely in science and engineering, there exist a lot of vague phenomena that do not behave with randomness. As a tool that attempts to deal with non-random phenomena, the concept of fuzzy set was initiated by Zadeh (1965) via membership function. To measure a fuzzy event, Zadeh (1978) proposed the concept of possibility measure. However, one disadvantage is that possibility measure is not self-dual and is not consistent with the law of excluded middle. Some people think that it is impossible for fuzzy mathematics to meet the law of excluded middle. In fact, as an improvement, Liu and Liu (2002) presented a Y. Liu (&) Department of Industrial Engineering, Tsinghua University, Beijing 100084, China e-mail: [email protected]
self-dual credibility measure for fuzzy event; then, it was consistent with the law of excluded middle. Based on possibility measure or credibility measure, a branch of mathematics has been developed by many scholars for studying the fuzzy phenomena. Fuzzy random variables are mathematical descriptions for fuzzy stochastic phenomena (i.e., a mixture of fuzziness and randomness) and are defined in several ways on the basis of probability theory and fuzzy mathematics. Kwakernaak (1978,1979) first introduced the notion of fuzzy random variable that is a function from a probability space to a collection of fuzzy variables. This concept was then developed by several researchers such as Puri and Ralescu (1986), Kruse and Meyer (1987) and Liu and Liu (2003) according to different requirements of measurability. The concept of chance measure of fuzzy random event was first given by Liu (2001a). After that, Liu and Liu (2005) proposed a concept of equilibrium chance measure. To rank fuzzy random variables, Liu and Liu (2003) presented a scalar expected value operator. In addition, different from Kwakernaak’s fuzzy random variable, Liu (2002, 2004) proposed a concept of random fuzzy variable that is a function from a possibility space to a collection of random variables. Although Zadeh’s fuzzy theory has been accepted and applied widely, it was still challenged by many scholars. A lot of surveys showed that human uncertainty does not behave like fuzziness. The focus of the debate is that the measure of union of events is not necessarily the maximum of measures of individual events. To model the information and knowledge such as ‘‘large’’, ‘‘warm’’, ‘‘young’’, ‘‘tall’’ and ‘‘most’’, an uncertainty theory was founded by Liu (2007) and refined by Liu (2010c). Nowadays, uncertainty theory has become a branch of mathematics for modeling human uncertainty.
123
626
Y. Liu
The paper assumes that human uncertainty and objective randomness simultaneously appear in a system. How do we model uncertain random phenomena? To answer this question, this paper will introduce the concepts of uncertain random variable, uncertain random arithmetic, chance measure, chance distribution, expected value operator and variance.
monotone increasing function except UðxÞ 0 and UðxÞ 1: The expected value of an uncertain variable n is defined by Liu (2007) as an average value of the uncertain variable in the sense of uncertain measure, i.e., E½n ¼
2 Preliminary
Axiom 1 (Normality Axiom) MfCg ¼ 1 for the universal set C: Axiom 2 (Duality Axiom) MfKg þ MfKc g ¼ 1 for any event K: Axiom 3 (Subadditivity Axiom) For every countable sequence of events K1 ; K2 ; . . .; we have ( ) 1 1 [ X M Ki MfKi g: i¼1
The triplet ðC; L; MÞ is called an uncertainty space. To obtain an uncertain measure of compound event, a product uncertain measure was defined by Liu (2009a), thus producing the fourth axiom of uncertainty theory: Axiom 4 (Product Axiom) Let ðCk ; Lk ; Mk Þ be uncertainty spaces for k ¼ 1; 2; . . .: The product uncertain measure M is an uncertain measure satisfying ( ) 1 1 Y ^ M Kk ¼ Mk fKk g k¼1
k¼1
where Kk are arbitrarily chosen events from Lk for k ¼ 1; 2; . . .; respectively. An uncertain variable (Liu 2007) is a measurable function n from an uncertainty space ðC; L; MÞ to the set of real numbers, i.e., for any Borel set B of real numbers, the set fn 2 Bg ¼ fc 2 C nðcÞ 2 Bg ð1Þ is an event. To describe an uncertain variable in practice, the concept of uncertainty distribution is defined by Liu (2007) as the following function UðxÞ ¼ Mfn xg:
ð2Þ
Peng and Iwamura (2010) proved that a function U : < ! [0, 1] is an uncertainty distribution if and only if it is a
123
Mfn rgdr
Z0
Mfn rgdr
ð3Þ
1
0
As a branch of axiomatic mathematics, the uncertainty theory was founded by Liu (2007) and refined by Liu (2010c). Let C be a nonempty set, and L a r-algebra over C: Each element K in L is called an event. A set function M from L to [0, 1] is called an uncertain measure if it satisfies the following axioms (Liu 2007):
i¼1
Zþ1
provided that at least one of the two integrals is finite. If n has an uncertainty distribution U, then the expected value may be calculated by E½n ¼
Zþ1
ð1 UðxÞÞdx
Z0 UðxÞdx:
ð4Þ
1
0
Let n1, n2,…, n3 be independent uncertain variables with uncertainty distributions U1 ; U2 ; . . .; Un , respectively. Liu (2010c) showed that if f ðx1 ; x2 ; . . .; xn Þ strictly increases with respect to x1 ; x2 ; . . .; xm and strictly decreaseses with respect to xmþ1 ; xmþ2 ; . . .; xn ; then n ¼ f ðn1 ; n2 ; . . .; nn Þ is an uncertain distribution
variable
with
inverse
uncertainty
1 W1 ðaÞ ¼ f ðU1 1 ðaÞ; . . .; Um ðaÞ; 1 1 Umþ1 ð1 aÞ; . . .; Un ð1 aÞÞ:
Furthermore, Liu and Ha (2010) proved that the uncertain variable n ¼ f ðn1 ; n2 ; . . .; nn Þ has an expected value E½n ¼
Z1
1 f ðU1 1 ðaÞ; . . .; Um ðaÞ;
0 1 U1 mþ1 ð1 aÞ; . . .; Un ð1 aÞÞda:
In what situations does uncertainty arise? Liu (2012a) wrote that ‘‘when the sample size is too small (even nosample) to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur. Since human beings usually overweight unlikely events, the belief degree may have much larger variance than the real frequency and then probability theory is no longer valid. In this situation, we should deal with it by uncertainty theory.’’ It may lead to counterintuitive results if we deal with the belief degree by probability theory. Uncertainty theory was applied widely. Liu (2009b) proposed a spectrum of uncertain programming that is a type of mathematical programming involving uncertain variables, and employed uncertain programming to model scheduling and logistics. In addition, uncertainty theory was also applied to uncertain statistics (Liu 2010c),
Uncertain random variables
uncertain risk analysis and uncertain reliability analysis (Liu 2010b), uncertain set (Liu 2010a, b), uncertain logic (Liu 2011), uncertain inference (Liu 2010a; Gao et al. 2010), uncertain calculus (Liu 2009a; Yao 2012), uncertain differential equation (Liu 2008; Chen and Liu 2010), uncertain control (Liu 2010a; Zhu 2010) and uncertain finance (Liu 2009a; Peng and Yao 2011; Liu and Ha 2009). For exploring the recent developments of uncertainty theory, the interested readers may consult Liu’s book Uncertainty Theory available at http://orsc.edu.cn/ liu/ut.pdf.
627
nðxÞ ¼ gðxÞu;
8x 2 X
is also an uncertain random variable provided that MfnðxÞ 2 Bg is a measurable function of x for any Borel set B of > : um ; if x ¼ xm is clearly an uncertain random variable.
is a measurable function of x from the probability space ðX; A; PrÞ to [0, 1]. Thus, MfnðxÞ 2 Bg is a random variable. Theorem 2 Let n be an uncertain random variable. If the expected value E[n(x)] is finite for each x, then E[n(x)] is a random variable. Proof To prove that the expected value E[n(x)] is a random variable, we only need to show that E[n(x)] is a measurable function of x. It is obvious that Zþ1 Z0 E½nðxÞ ¼ MfnðxÞ rgdr MfnðxÞ rgdr 1
0
j lj M nðxÞ ¼ lim lim j!1 k!1 k k l¼1 ! k X j lj M nðxÞ k k l¼1 k X
Since MfnðxÞ lj=kg and MfnðxÞ lj=kg are all measurable functions for any integers j, k and l, the expected value E[n(x)] is a measurable function of x. The proof is complete. Definition 2 (Uncertain random arithmetic) Let f be a measurable function, and n1 ; n2 ; . . .; nn uncertain random variables on the probability space ðX; A; PrÞ; i ¼ 1; 2; . . .; n: Then, n ¼ f ðn1 ; n2 ; . . .; nn Þ is an uncertain random variable defined by nðxÞ ¼ f ðn1 ðxÞ; n2 ðxÞ; . . .; nn ðxÞÞ
ð5Þ
in the sense of operations of uncertain variables.
Example 4 If g is a random variable defined on the probability space ðX; A; PrÞ; and u is an uncertain variable, then the sum n = g ? u is an uncertain random variable defined by
Example 5 Let n1 and n2 be two uncertain random variables defined on the probability space ðX; A; PrÞ: Then the sum n = n1 ? n2 is an uncertain random variable defined by
nðxÞ ¼ gðxÞ þ u;
nðxÞ ¼ n1 ðxÞ þ n2 ðxÞ:
8x 2 X
provided that MfnðxÞ 2 Bg is a measurable function of x for any Borel set B of < s2 with probability p2 n¼ ... > > : sn with probability pn :
123
Chfn 2 0:2x þ 0:2; if 3\x 4 > > : 1; if 4 [ x:
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Y. Liu
Example 13 Let s1 ; s2 ; . . .; sn be uncertain variables with uncertainty distributions !1 ; !2 ; . . .; !n ; respectively, and let us define an uncertain random variable 8 s1 with probability p1 > > < s2 with probability p2 n¼ ... > > : sn with probability pn :
Thus for almost all x 2 X; we have MfnðxÞ xg 0;
8x 2
< s2 with probability p2 ð12Þ n¼ ... > > : sn with probability pn
Zþ1
Chfn þ b rgdr
0
¼
Zþ1
Chfn r bgdr
Z0 1 Z0
Chfn þ b rgdr
Chfn r bgdr
1
0
¼ E½n þ
Zb
ðChfn r bg þ Chfn\r bgÞdr
0
¼ E½n þ b:
has an expected value Zþ1
E½n ¼
ð1 UðxÞÞdx
0 Zþ1
¼
Z0
If b \ 0, then we have
UðxÞdx 1
1
n X
!
pi !i ðxÞ dx
Z0 X n
i¼1
E½n þ b ¼ E½n pi !i ðxÞdx
¼
i¼1 n X
¼ E½n þ b: Furthermore, if a = 0, then E[an] = aE[n] holds trivially. If a [ 0, we have
1
0
pi E½si : E½an ¼
i¼1
Zþ1
n X
pi E½si :
ð13Þ
¼a
Theorem 9 Let n be an uncertain random variable with chance distribution U: If the expected value exists, then Z1
ð14Þ
E½n ¼
Zþ1
ð1 UðxÞÞdx
¼
Uð0Þ
Z0
U1 ðaÞda þ
ZUð0Þ 0
U1 ðaÞda ¼
Chfn tgdt ¼ aE½n:
Chfan rgdr
Chfan rgdr
1
0
¼ a
Z0
Z0
Chfn rgdr þ a
1
Zþ1
Chfn rgdr
0
¼ aE½n: The theorem is thus proved.
UðxÞdx 1
0
Z1
Chfn tgdt a
Z0 1
Zþ1
0
Proof It follows from the definitions of expected value operator and chance distribution that
Chfan rgdr
If a \ 0, we have E½an ¼
U1 ðaÞda:
Z0 1
Zþ1 0
i¼1
E½n ¼
Chfan rgdr
0
That is, E½n ¼
ðChfn r bg þ Chfn\r bgÞdr
b
i¼1
1 0 þ1 1 0 Z Z n X ¼ pi @ ð1 !i ðxÞÞdx !i ðxÞdxA 0
Z0
Z1
U1 ðaÞda:
0
The theorem is thus proved. Theorem 10 Let n be an uncertain random variable whose expected value exists. Then for any real numbers a and b, we have
Theorem 11 Let n be an uncertain random variable, and f a nonnegative function. If f is even and increasing on ½0; 1Þ; then for any given number t [ 0, we have Chfjnj tg
E½f ðnÞ : f ðtÞ
ð16Þ
Proof Since Ch{|n| C f-1(r)} is a monotone decreasing function of r on ½0; 1Þ; it follows from the nonnegativity of f(n) that
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Y. Liu
E½f ðnÞ ¼
¼
Zþ1 0 Zþ1
Chfðn eÞ2 rg ¼ 0; Chff ðnÞ rgdr
8r [ 0
and Chfðn eÞ2 ¼ 0g ¼ 1:
Chfjnj f 1 ðrÞgdr
Hence,
0
Z
f ðtÞ
Chfn ¼ eg ¼ 1:
Chfjnj f 1 ðrÞgdr
Conversely, assume Ch{n = e} = 1. Then we have
0
Z
f ðtÞ
dr Chfjnj f 1 ðf ðtÞÞg ¼ f ðtÞ Chfjnj tg:
Chfðn eÞ2 ¼ 0g ¼ 1:
0
It follows from Theorem 5 that The theorem is thus proved.
Chfðn eÞ2 rg ¼ 0;
Theorem 12 Let n be an uncertain random variable. Then for any given numbers t [ 0 and p [ 0, we have Chfjnj tg Proof (17).
E½jnjp : tp
ð17Þ
Taking f(x) = |x|p and applying (15), we obtain
Hence, V½n ¼
Zþ1
Chfðe eÞ2 rgdr ¼ 0:
0
The theorem is proved.
7 Variance Definition 6 Let n be an uncertain random variable with finite expected value e. Then the variance of n is defined by V[n] = E[(n - e)2]. Since (n - e)2 is a nonnegative uncertain random variable, we immediately have V½n ¼
8r [ 0:
Zþ1
Chfðn eÞ2 rgdr:
Theorem 15 Let n be an uncertain random variable whose variance V[n] exists. Then for any given number t [ 0, we have Chfjn E½nj tg
V½n : t2
ð18Þ
Proof It follows from Theorem*11 immediately when the uncertain random variable n is replaced with n E[n], and f(x) = x2.
8 Random uncertain variables
0
Theorem 13 If n is an uncertain random variable with finite expected value, a and b are real numbers, then V[an ? b] = a2V[n]. Proof Let e be the expected value of n. Then, an ? b has an expected value ae ? b. Thus, the variance is
Different from uncertain random variable, we may also define a random uncertain variable as a measurable function from an uncertainty space to the set of random variables.
V½an þ b ¼ E½ððan þ bÞ ðae þ bÞÞ2 ¼ E½a2 ðn eÞ2 ¼ a2 E½ðn eÞ2 ¼ a2 V½n:
Definition 7 A random uncertain variable is a function n from an uncertainty space ðC; L; MÞ to the set of random variables such that PrfnðcÞ 2 Bg is a measurable function of c for any Borel set B of
FOCUS
Uncertain random variables: a mixture of uncertainty and randomness Yuhan Liu
Published online: 27 September 2012 Springer-Verlag 2012
Abstract In many cases, human uncertainty and objective randomness simultaneously appear in a system. In order to describe this phenomena, this paper presents a new concept of uncertain random variable. To measure uncertain random events, this paper also combines probability measure and uncertain measure into a chance measure. Based on the tool of chance measure, the concepts of chance distribution, expected value and variance of uncertain random variable are proposed. Keywords Uncertainty theory Probability theory Uncertain random variable Chance measure
1 Introduction Probability theory (Kolmogorov 1933) is a branch of mathematics for studying the behavior of random phenomena. Although probability theory has been applied widely in science and engineering, there exist a lot of vague phenomena that do not behave with randomness. As a tool that attempts to deal with non-random phenomena, the concept of fuzzy set was initiated by Zadeh (1965) via membership function. To measure a fuzzy event, Zadeh (1978) proposed the concept of possibility measure. However, one disadvantage is that possibility measure is not self-dual and is not consistent with the law of excluded middle. Some people think that it is impossible for fuzzy mathematics to meet the law of excluded middle. In fact, as an improvement, Liu and Liu (2002) presented a Y. Liu (&) Department of Industrial Engineering, Tsinghua University, Beijing 100084, China e-mail: [email protected]
self-dual credibility measure for fuzzy event; then, it was consistent with the law of excluded middle. Based on possibility measure or credibility measure, a branch of mathematics has been developed by many scholars for studying the fuzzy phenomena. Fuzzy random variables are mathematical descriptions for fuzzy stochastic phenomena (i.e., a mixture of fuzziness and randomness) and are defined in several ways on the basis of probability theory and fuzzy mathematics. Kwakernaak (1978,1979) first introduced the notion of fuzzy random variable that is a function from a probability space to a collection of fuzzy variables. This concept was then developed by several researchers such as Puri and Ralescu (1986), Kruse and Meyer (1987) and Liu and Liu (2003) according to different requirements of measurability. The concept of chance measure of fuzzy random event was first given by Liu (2001a). After that, Liu and Liu (2005) proposed a concept of equilibrium chance measure. To rank fuzzy random variables, Liu and Liu (2003) presented a scalar expected value operator. In addition, different from Kwakernaak’s fuzzy random variable, Liu (2002, 2004) proposed a concept of random fuzzy variable that is a function from a possibility space to a collection of random variables. Although Zadeh’s fuzzy theory has been accepted and applied widely, it was still challenged by many scholars. A lot of surveys showed that human uncertainty does not behave like fuzziness. The focus of the debate is that the measure of union of events is not necessarily the maximum of measures of individual events. To model the information and knowledge such as ‘‘large’’, ‘‘warm’’, ‘‘young’’, ‘‘tall’’ and ‘‘most’’, an uncertainty theory was founded by Liu (2007) and refined by Liu (2010c). Nowadays, uncertainty theory has become a branch of mathematics for modeling human uncertainty.
123
626
Y. Liu
The paper assumes that human uncertainty and objective randomness simultaneously appear in a system. How do we model uncertain random phenomena? To answer this question, this paper will introduce the concepts of uncertain random variable, uncertain random arithmetic, chance measure, chance distribution, expected value operator and variance.
monotone increasing function except UðxÞ 0 and UðxÞ 1: The expected value of an uncertain variable n is defined by Liu (2007) as an average value of the uncertain variable in the sense of uncertain measure, i.e., E½n ¼
2 Preliminary
Axiom 1 (Normality Axiom) MfCg ¼ 1 for the universal set C: Axiom 2 (Duality Axiom) MfKg þ MfKc g ¼ 1 for any event K: Axiom 3 (Subadditivity Axiom) For every countable sequence of events K1 ; K2 ; . . .; we have ( ) 1 1 [ X M Ki MfKi g: i¼1
The triplet ðC; L; MÞ is called an uncertainty space. To obtain an uncertain measure of compound event, a product uncertain measure was defined by Liu (2009a), thus producing the fourth axiom of uncertainty theory: Axiom 4 (Product Axiom) Let ðCk ; Lk ; Mk Þ be uncertainty spaces for k ¼ 1; 2; . . .: The product uncertain measure M is an uncertain measure satisfying ( ) 1 1 Y ^ M Kk ¼ Mk fKk g k¼1
k¼1
where Kk are arbitrarily chosen events from Lk for k ¼ 1; 2; . . .; respectively. An uncertain variable (Liu 2007) is a measurable function n from an uncertainty space ðC; L; MÞ to the set of real numbers, i.e., for any Borel set B of real numbers, the set fn 2 Bg ¼ fc 2 C nðcÞ 2 Bg ð1Þ is an event. To describe an uncertain variable in practice, the concept of uncertainty distribution is defined by Liu (2007) as the following function UðxÞ ¼ Mfn xg:
ð2Þ
Peng and Iwamura (2010) proved that a function U : < ! [0, 1] is an uncertainty distribution if and only if it is a
123
Mfn rgdr
Z0
Mfn rgdr
ð3Þ
1
0
As a branch of axiomatic mathematics, the uncertainty theory was founded by Liu (2007) and refined by Liu (2010c). Let C be a nonempty set, and L a r-algebra over C: Each element K in L is called an event. A set function M from L to [0, 1] is called an uncertain measure if it satisfies the following axioms (Liu 2007):
i¼1
Zþ1
provided that at least one of the two integrals is finite. If n has an uncertainty distribution U, then the expected value may be calculated by E½n ¼
Zþ1
ð1 UðxÞÞdx
Z0 UðxÞdx:
ð4Þ
1
0
Let n1, n2,…, n3 be independent uncertain variables with uncertainty distributions U1 ; U2 ; . . .; Un , respectively. Liu (2010c) showed that if f ðx1 ; x2 ; . . .; xn Þ strictly increases with respect to x1 ; x2 ; . . .; xm and strictly decreaseses with respect to xmþ1 ; xmþ2 ; . . .; xn ; then n ¼ f ðn1 ; n2 ; . . .; nn Þ is an uncertain distribution
variable
with
inverse
uncertainty
1 W1 ðaÞ ¼ f ðU1 1 ðaÞ; . . .; Um ðaÞ; 1 1 Umþ1 ð1 aÞ; . . .; Un ð1 aÞÞ:
Furthermore, Liu and Ha (2010) proved that the uncertain variable n ¼ f ðn1 ; n2 ; . . .; nn Þ has an expected value E½n ¼
Z1
1 f ðU1 1 ðaÞ; . . .; Um ðaÞ;
0 1 U1 mþ1 ð1 aÞ; . . .; Un ð1 aÞÞda:
In what situations does uncertainty arise? Liu (2012a) wrote that ‘‘when the sample size is too small (even nosample) to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur. Since human beings usually overweight unlikely events, the belief degree may have much larger variance than the real frequency and then probability theory is no longer valid. In this situation, we should deal with it by uncertainty theory.’’ It may lead to counterintuitive results if we deal with the belief degree by probability theory. Uncertainty theory was applied widely. Liu (2009b) proposed a spectrum of uncertain programming that is a type of mathematical programming involving uncertain variables, and employed uncertain programming to model scheduling and logistics. In addition, uncertainty theory was also applied to uncertain statistics (Liu 2010c),
Uncertain random variables
uncertain risk analysis and uncertain reliability analysis (Liu 2010b), uncertain set (Liu 2010a, b), uncertain logic (Liu 2011), uncertain inference (Liu 2010a; Gao et al. 2010), uncertain calculus (Liu 2009a; Yao 2012), uncertain differential equation (Liu 2008; Chen and Liu 2010), uncertain control (Liu 2010a; Zhu 2010) and uncertain finance (Liu 2009a; Peng and Yao 2011; Liu and Ha 2009). For exploring the recent developments of uncertainty theory, the interested readers may consult Liu’s book Uncertainty Theory available at http://orsc.edu.cn/ liu/ut.pdf.
627
nðxÞ ¼ gðxÞu;
8x 2 X
is also an uncertain random variable provided that MfnðxÞ 2 Bg is a measurable function of x for any Borel set B of > : um ; if x ¼ xm is clearly an uncertain random variable.
is a measurable function of x from the probability space ðX; A; PrÞ to [0, 1]. Thus, MfnðxÞ 2 Bg is a random variable. Theorem 2 Let n be an uncertain random variable. If the expected value E[n(x)] is finite for each x, then E[n(x)] is a random variable. Proof To prove that the expected value E[n(x)] is a random variable, we only need to show that E[n(x)] is a measurable function of x. It is obvious that Zþ1 Z0 E½nðxÞ ¼ MfnðxÞ rgdr MfnðxÞ rgdr 1
0
j lj M nðxÞ ¼ lim lim j!1 k!1 k k l¼1 ! k X j lj M nðxÞ k k l¼1 k X
Since MfnðxÞ lj=kg and MfnðxÞ lj=kg are all measurable functions for any integers j, k and l, the expected value E[n(x)] is a measurable function of x. The proof is complete. Definition 2 (Uncertain random arithmetic) Let f be a measurable function, and n1 ; n2 ; . . .; nn uncertain random variables on the probability space ðX; A; PrÞ; i ¼ 1; 2; . . .; n: Then, n ¼ f ðn1 ; n2 ; . . .; nn Þ is an uncertain random variable defined by nðxÞ ¼ f ðn1 ðxÞ; n2 ðxÞ; . . .; nn ðxÞÞ
ð5Þ
in the sense of operations of uncertain variables.
Example 4 If g is a random variable defined on the probability space ðX; A; PrÞ; and u is an uncertain variable, then the sum n = g ? u is an uncertain random variable defined by
Example 5 Let n1 and n2 be two uncertain random variables defined on the probability space ðX; A; PrÞ: Then the sum n = n1 ? n2 is an uncertain random variable defined by
nðxÞ ¼ gðxÞ þ u;
nðxÞ ¼ n1 ðxÞ þ n2 ðxÞ:
8x 2 X
provided that MfnðxÞ 2 Bg is a measurable function of x for any Borel set B of < s2 with probability p2 n¼ ... > > : sn with probability pn :
123
Chfn 2 0:2x þ 0:2; if 3\x 4 > > : 1; if 4 [ x:
123
630
Y. Liu
Example 13 Let s1 ; s2 ; . . .; sn be uncertain variables with uncertainty distributions !1 ; !2 ; . . .; !n ; respectively, and let us define an uncertain random variable 8 s1 with probability p1 > > < s2 with probability p2 n¼ ... > > : sn with probability pn :
Thus for almost all x 2 X; we have MfnðxÞ xg 0;
8x 2
< s2 with probability p2 ð12Þ n¼ ... > > : sn with probability pn
Zþ1
Chfn þ b rgdr
0
¼
Zþ1
Chfn r bgdr
Z0 1 Z0
Chfn þ b rgdr
Chfn r bgdr
1
0
¼ E½n þ
Zb
ðChfn r bg þ Chfn\r bgÞdr
0
¼ E½n þ b:
has an expected value Zþ1
E½n ¼
ð1 UðxÞÞdx
0 Zþ1
¼
Z0
If b \ 0, then we have
UðxÞdx 1
1
n X
!
pi !i ðxÞ dx
Z0 X n
i¼1
E½n þ b ¼ E½n pi !i ðxÞdx
¼
i¼1 n X
¼ E½n þ b: Furthermore, if a = 0, then E[an] = aE[n] holds trivially. If a [ 0, we have
1
0
pi E½si : E½an ¼
i¼1
Zþ1
n X
pi E½si :
ð13Þ
¼a
Theorem 9 Let n be an uncertain random variable with chance distribution U: If the expected value exists, then Z1
ð14Þ
E½n ¼
Zþ1
ð1 UðxÞÞdx
¼
Uð0Þ
Z0
U1 ðaÞda þ
ZUð0Þ 0
U1 ðaÞda ¼
Chfn tgdt ¼ aE½n:
Chfan rgdr
Chfan rgdr
1
0
¼ a
Z0
Z0
Chfn rgdr þ a
1
Zþ1
Chfn rgdr
0
¼ aE½n: The theorem is thus proved.
UðxÞdx 1
0
Z1
Chfn tgdt a
Z0 1
Zþ1
0
Proof It follows from the definitions of expected value operator and chance distribution that
Chfan rgdr
If a \ 0, we have E½an ¼
U1 ðaÞda:
Z0 1
Zþ1 0
i¼1
E½n ¼
Chfan rgdr
0
That is, E½n ¼
ðChfn r bg þ Chfn\r bgÞdr
b
i¼1
1 0 þ1 1 0 Z Z n X ¼ pi @ ð1 !i ðxÞÞdx !i ðxÞdxA 0
Z0
Z1
U1 ðaÞda:
0
The theorem is thus proved. Theorem 10 Let n be an uncertain random variable whose expected value exists. Then for any real numbers a and b, we have
Theorem 11 Let n be an uncertain random variable, and f a nonnegative function. If f is even and increasing on ½0; 1Þ; then for any given number t [ 0, we have Chfjnj tg
E½f ðnÞ : f ðtÞ
ð16Þ
Proof Since Ch{|n| C f-1(r)} is a monotone decreasing function of r on ½0; 1Þ; it follows from the nonnegativity of f(n) that
123
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Y. Liu
E½f ðnÞ ¼
¼
Zþ1 0 Zþ1
Chfðn eÞ2 rg ¼ 0; Chff ðnÞ rgdr
8r [ 0
and Chfðn eÞ2 ¼ 0g ¼ 1:
Chfjnj f 1 ðrÞgdr
Hence,
0
Z
f ðtÞ
Chfn ¼ eg ¼ 1:
Chfjnj f 1 ðrÞgdr
Conversely, assume Ch{n = e} = 1. Then we have
0
Z
f ðtÞ
dr Chfjnj f 1 ðf ðtÞÞg ¼ f ðtÞ Chfjnj tg:
Chfðn eÞ2 ¼ 0g ¼ 1:
0
It follows from Theorem 5 that The theorem is thus proved.
Chfðn eÞ2 rg ¼ 0;
Theorem 12 Let n be an uncertain random variable. Then for any given numbers t [ 0 and p [ 0, we have Chfjnj tg Proof (17).
E½jnjp : tp
ð17Þ
Taking f(x) = |x|p and applying (15), we obtain
Hence, V½n ¼
Zþ1
Chfðe eÞ2 rgdr ¼ 0:
0
The theorem is proved.
7 Variance Definition 6 Let n be an uncertain random variable with finite expected value e. Then the variance of n is defined by V[n] = E[(n - e)2]. Since (n - e)2 is a nonnegative uncertain random variable, we immediately have V½n ¼
8r [ 0:
Zþ1
Chfðn eÞ2 rgdr:
Theorem 15 Let n be an uncertain random variable whose variance V[n] exists. Then for any given number t [ 0, we have Chfjn E½nj tg
V½n : t2
ð18Þ
Proof It follows from Theorem*11 immediately when the uncertain random variable n is replaced with n E[n], and f(x) = x2.
8 Random uncertain variables
0
Theorem 13 If n is an uncertain random variable with finite expected value, a and b are real numbers, then V[an ? b] = a2V[n]. Proof Let e be the expected value of n. Then, an ? b has an expected value ae ? b. Thus, the variance is
Different from uncertain random variable, we may also define a random uncertain variable as a measurable function from an uncertainty space to the set of random variables.
V½an þ b ¼ E½ððan þ bÞ ðae þ bÞÞ2 ¼ E½a2 ðn eÞ2 ¼ a2 E½ðn eÞ2 ¼ a2 V½n:
Definition 7 A random uncertain variable is a function n from an uncertainty space ðC; L; MÞ to the set of random variables such that PrfnðcÞ 2 Bg is a measurable function of c for any Borel set B of
