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| {{distinguish|Uncertainty principle}}
| | If you are looking for a specific plugin, then you can just search for the name of the plugin. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. If you have any concerns about exactly where and how to use [https://aidl.org/wordpress_dropbox_backup_457720 wordpress backup], you can get hold of us at our own web site. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. In the most current edition you can customize your retailer layout and display hues and fonts similar to your site or blog. <br><br>Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. Some of the Wordpress development services offered by us are:. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. s cutthroat competition prevailing in the online space won. We can active Akismet from wp-admin > Plugins > Installed Plugins. Provide the best and updated information to the web searchers and make use of these wonderful free themes and create beautiful websites. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. An ease of use which pertains to both internet site back-end and front-end users alike. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Page speed is an important factor in ranking, especially with Google. For your information, it is an open source web content management system. |
| {{multiple issues|
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| {{one source|date=November 2009}}
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| {{jargon|date=December 2009}}
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| {{howto|date=December 2009}}
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| {{citation style|date=December 2009}}
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| {{misleading|date=December 2009}}
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| }}
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| '''Uncertainty theory''' is a branch of [[mathematics]] based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.{{Clarify|date=December 2009}} It was founded by Baoding Liu <ref>Baoding Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.</ref> in 2007 and refined in 2009.<ref>Baoding Liu, Uncertainty Theory, 4th ed., http://orsc.edu.cn/liu/ut.pdf.</ref>
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| Mathematical measures of the likelihood of an event being true include [[probability theory]], capacity, [[fuzzy logic]], possibility, and credibility, as well as uncertainty.
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| ==Five axioms==
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| '''Axiom 1.''' (Normality Axiom) <math>\mathcal{M}\{\Gamma\}=1\text{ for the universal set }\Gamma</math>.
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| '''Axiom 2.''' (Monotonicity Axiom) <math>\mathcal{M}\{\Lambda_1\}\le\mathcal{M}\{\Lambda_2\}\text{ whenever }\Lambda_1\subset\Lambda_2</math>.
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| '''Axiom 3.''' (Self-Duality Axiom) <Math>\mathcal{M}\{\Lambda\}+\mathcal{M}\{\Lambda^c\}=1\text{ for any event }\Lambda</math>.
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| '''Axiom 4.''' (Countable Subadditivity Axiom) For every countable sequence of events Λ<sub>1</sub>, Λ<sub>2</sub>, ..., we have
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| ::<math>\mathcal{M}\left\{\bigcup_{i=1}^\infty\Lambda_i\right\}\le\sum_{i=1}^\infty\mathcal{M}\{\Lambda_i\}</math>.
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| '''Axiom 5.''' (Product Measure Axiom) Let <math>(\Gamma_k,\mathcal{L}_k,\mathcal{M}_k)</math> be uncertainty spaces for <math>k=1,2,\cdots,n</math>. Then the product uncertain measure <math>\mathcal{M}</math> is an uncertain measure on the product σ-algebra satisfying
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| ::<math>\mathcal{M}\left\{\prod_{i=1}^n\Lambda_i\right\}=\underset{1\le i\le n}{\operatorname{min} }\mathcal{M}_i\{\Lambda_i\}</math>.
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| '''Principle.''' (Maximum Uncertainty Principle) For any event, if there are multiple reasonable values that an uncertain measure may take, then the value as close to 0.5 as possible is assigned to the event.
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| ==Uncertain variables==
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| An uncertain variable is a [[measurable function]] ξ from an uncertainty space <math>(\Gamma,L,M)</math> to the [[set (mathematics)|set]] of [[real numbers]], i.e., for any [[Borel set]] '''B''' of [[real numbers]], the set
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| <math>\{\xi\in B\}=\{\gamma \in \Gamma|\xi(\gamma)\in B\}</math> is an event.
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| ==Uncertainty distribution==
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| Uncertainty distribution is inducted to describe uncertain variables.
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| '''Definition''':The '''uncertainty distribution''' <math>\Phi(x):R \rightarrow [0,1]</math> of an uncertain variable ξ is defined by <math>\Phi(x)=M\{\xi\leq x\}</math>.
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| '''Theorem'''(Peng and Iwamura, ''Sufficient and Necessary Condition for Uncertainty Distribution'') A function <math>\Phi(x):R \rightarrow [0,1]</math> is an uncertain distribution if and only if it is an increasing function except <math>\Phi (x) \equiv 0</math> and <math>\Phi (x)\equiv 1</math>.
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| ==Independence==
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| '''Definition''': The uncertain variables <math>\xi_1,\xi_2,\ldots,\xi_m</math> are said to be independent if
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| :<math>M\{\cap_{i=1}^m(\xi \in B_i)\}=\mbox{min}_{1\leq i \leq m}M\{\xi_i \in B_i\} </math>
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| for any Borel sets <math>B_1,B_2,\ldots,B_m</math> of real numbers.
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| '''Theorem 1''': The uncertain variables <math>\xi_1,\xi_2,\ldots,\xi_m</math> are independent if
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| :<math>M\{\cup_{i=1}^m(\xi \in B_i)\}=\mbox{max}_{1\leq i \leq m}M\{\xi_i \in B_i\} </math>
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| for any Borel sets <math>B_1,B_2,\ldots,B_m</math> of real numbers.
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| '''Theorem 2''': Let <math>\xi_1,\xi_2,\ldots,\xi_m</math> be independent uncertain variables, and <math>f_1,f_2,\ldots,f_m</math> measurable functions. Then <math>f_1(\xi_1),f_2(\xi_2),\ldots,f_m(\xi_m)</math> are independent uncertain variables.
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| '''Theorem 3''': Let <math>\Phi_i</math> be uncertainty distributions of independent uncertain variables <math>\xi_i,\quad i=1,2,\ldots,m</math> respectively, and <math>\Phi</math> the joint uncertainty distribution of uncertain vector <math>(\xi_1,\xi_2,\ldots,\xi_m)</math>. If <math>\xi_1,\xi_2,\ldots,\xi_m</math> are independent, then we have
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| :<math>\Phi(x_1, x_2, \ldots, x_m)=\mbox{min}_{1\leq i \leq m}\Phi_i(x_i)</math>
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| for any real numbers <math>x_1, x_2, \ldots, x_m</math>.
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| ==Operational law==
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| '''Theorem''': Let <math>\xi_1,\xi_2,\ldots,\xi_m</math> be independent uncertain variables, and <math>f: R^n \rightarrow R</math> a measurable function. Then <math>\xi=f(\xi_1,\xi_2,\ldots,\xi_m)</math> is an uncertain variable such that
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| ::<math>\mathcal{M}\{\xi\in B\}=\begin{cases} \underset{f(B_1,B_2,\cdots,B_n)\subset B}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\}, & \text{if } \underset{f(B_1,B_2,\cdots,B_n)\subset B}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\} > 0.5 \\ 1-\underset{f(B_1,B_2,\cdots,B_n)\subset B^c}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\}, & \text{if } \underset{f(B_1,B_2,\cdots,B_n)\subset B^c}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\} > 0.5 \\ 0.5, & \text{otherwise} \end{cases}</math>
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| where <math>B, B_1, B_2, \ldots, B_m</math> are Borel sets, and <math>f( B_1, B_2, \ldots, B_m)\subset B</math> means<math>f(x_1, x_2, \ldots, x_m) \in B</math> for any<math>x_1 \in B_1, x_2 \in B_2, \ldots,x_m \in B_m</math>.
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| ==Expected Value==
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| '''Definition''': Let <math>\xi</math> be an uncertain variable. Then the expected value of <math>\xi</math> is defined by
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| :::<math>E[\xi]=\int_0^{+\infty}M\{\xi\geq r\}dr-\int_{-\infty}^0M\{\xi\leq r\}dr</math>
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| provided that at least one of the two integrals is finite.
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| '''Theorem 1''': Let <math>\xi</math> be an uncertain variable with uncertainty distribution <math>\Phi</math>. If the expected value exists, then
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| :::<math>E[\xi]=\int_0^{+\infty}(1-\Phi(x))dx-\int_{-\infty}^0\Phi(x)dx</math>.
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| [[File:Uncertain expected value.jpg|300px|center]]
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| '''Theorem 2''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi</math>. If the expected value exists, then
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| :::<math>E[\xi]=\int_0^1\Phi^{-1}(\alpha)d\alpha</math>.
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| '''Theorem 3''': Let <math>\xi</math> and <math>\eta</math> be independent uncertain variables with finite expected values. Then for any real numbers <math>a</math> and <math>b</math>, we have
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| :::<math>E[a\xi+b\eta]=aE[\xi]+b[\eta]</math>.
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| ==Variance==
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| '''Definition''': Let <math>\xi</math> be an uncertain variable with finite expected value <math>e</math>. Then the variance of <math>\xi</math> is defined by
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| :::<math>V[\xi]=E[(\xi-e)^2]</math>.
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| '''Theorem''': If <math>\xi</math> be an uncertain variable with finite expected value, <math>a</math> and <math>b</math> are real numbers, then
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| :::<math>V[a\xi+b]=a^2V[\xi]</math>.
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| ==Critical value==
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| '''Definition''': Let <math>\xi</math> be an uncertain variable, and <math>\alpha\in(0,1]</math>. Then
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| :<math>\xi_{sup}(\alpha)=\mbox{sup}\{r|M\{\xi\geq r\}\geq\alpha\}</math>
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| is called the α-[[optimistic]] value to <math>\xi</math>, and | |
| :<math>\xi_{inf}(\alpha)=\mbox{inf}\{r|M\{\xi\leq r\}\geq\alpha\}</math>
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| is called the α-[[pessimistic]] value to <math>\xi</math>.
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| '''Theorem 1''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi</math>. Then its α-[[optimistic]] value and α-[[pessimistic]] value are
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| ::<math>\xi_{sup}(\alpha)=\Phi^{-1}(1-\alpha)</math>,
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| ::<math>\xi_{inf}(\alpha)=\Phi^{-1}(\alpha)</math>.
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| '''Theorem 2''': Let <math>\xi</math> be an uncertain variable, and <math>\alpha\in(0,1]</math>. Then we have
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| * if <math>\alpha>0.5</math>, then <math>\xi_{inf}(\alpha)\geq \xi_{sup}(\alpha)</math>;
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| * if <math>\alpha\leq 0.5</math>, then <math>\xi_{inf}(\alpha)\leq \xi_{sup}(\alpha)</math>.
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| '''Theorem 3''': Suppose that <math>\xi</math> and <math>\eta</math> are independent uncertain variables, and <math>\alpha\in(0,1]</math>. Then we have
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| <math>(\xi + \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)+\eta_{sup}{\alpha}</math>,
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| <math>(\xi + \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)+\eta_{inf}{\alpha}</math>,
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| <math>(\xi \vee \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\vee\eta_{sup}{\alpha}</math>,
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| <math>(\xi \vee \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\vee\eta_{inf}{\alpha}</math>,
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| <math>(\xi \wedge \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\wedge\eta_{sup}{\alpha}</math>,
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| <math>(\xi \wedge \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\wedge\eta_{inf}{\alpha}</math>.
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| ==Entropy==
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| '''Definition''': Let <math>\xi</math> be an uncertain variable with uncertainty distribution <math>\Phi</math>. Then its entropy is defined by
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| ::<math>H[\xi]=\int_{-\infty}^{+\infty}S(\Phi(x))dx</math>
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| where <math>S(x)=-t\mbox{ln}(t)-(1-t)\mbox{ln}(1-t)</math>.
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| '''Theorem 1'''(''Dai and Chen''): Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi</math>. Then
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| ::<math>H[\xi]=\int_0^1\Phi^{-1}(\alpha)\mbox{ln}\frac{\alpha}{1-\alpha}d\alpha</math>.
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| '''Theorem 2''': Let <math>\xi</math> and <math>\eta</math> be independent uncertain variables. Then for any real numbers <math>a</math> and <math>b</math>, we have
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| ::<math>H[a\xi+b\eta]=|a|E[\xi]+|b|E[\eta]</math>.
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| '''Theorem 3''': Let <math>\xi</math> be an uncertain variable whose uncertainty distribution is arbitrary but the expected value <math>e</math> and variance <math>\sigma^2</math>. Then
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| ::<math>H[\xi]\leq\frac{\pi\sigma}{\sqrt{3}}</math>.
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| ==Inequalities==
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| '''Theorem 1'''(''Liu'', Markov Inequality): Let <math>\xi</math> be an uncertain variable. Then for any given numbers <math>t > 0</math> and <math>p > 0</math>, we have
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| ::<math>M\{|\xi|\geq t\}\leq \frac{E[|\xi|^p]}{t^p}</math>.
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| '''Theorem 2''' (''Liu'', Chebyshev Inequality) Let <math>\xi</math> be an uncertain variable whose variance <math>V[\xi]</math> exists. Then for any given number<math> t > 0</math>, we have
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| ::<math>M\{|\xi-E[\xi]|\geq t\}\leq \frac{V[\xi]}{t^2}</math>.
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| '''Theorem 3''' (''Liu'', Holder’s Inequality) Let <math>p</math> and <math>q</math> be positive numbers with <math>1/p + 1/q = 1</math>, and let <math>\xi</math> and <math>\eta</math> be independent uncertain variables with <math>E[|\xi|^p]< \infty</math> and <math>E[|\eta|^q] < \infty</math>. Then we have
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| ::<math>E[|\xi\eta|]\leq \sqrt[p]{E[|\xi|^p]} \sqrt[p]{E[\eta|^p]}</math>.
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| '''Theorem 4''':(Liu [127], Minkowski Inequality) Let <math>p</math> be a real number with <math>p\leq 1</math>, and let <math>\xi</math> and <math>\eta</math> be independent uncertain variables with <math>E[|\xi|^p]< \infty</math> and <math>E[|\eta|^q] < \infty</math>. Then we have | |
| ::<math>\sqrt[p]{E[|\xi+\eta|^p]}\leq \sqrt[p]{E[|\xi|^p]}+\sqrt[p]{E[\eta|^p]}</math>.
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| ==Convergence concept==
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| '''Definition 1''': Suppose that <math>\xi,\xi_1,\xi_2,\ldots</math> are uncertain variables defined on the uncertainty space <math>(\Gamma,L,M)</math>. The sequence <math>\{\xi_i\}</math> is said to be convergent a.s. to <math>\xi</math> if there exists an event <math>\Lambda</math> with <math>M\{\Lambda\} = 1</math> such that
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| ::<math>\mbox{lim}_{i\rightarrow\infty}|\xi_i(\gamma)-\xi(\gamma)|=0</math>
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| for every <math>\gamma\in\Lambda</math>. In that case we write <math>\xi_i\rightarrow \xi</math>,a.s.
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| '''Definition 2''': Suppose that <math>\xi,\xi_1,\xi_2,\ldots</math> are uncertain variables. We say that the sequence <math>\{\xi_i\}</math> converges in measure to <math>\xi</math> if
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| ::<math>\mbox{lim}_{i\rightarrow\infty}M\{|\xi_i-\xi|\leq \varepsilon \}=0</math>
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| for every <math>\varepsilon>0</math>.
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| '''Definition 3''': Suppose that <math>\xi,\xi_1,\xi_2,\ldots</math> are uncertain variables with finite expected values. We say that the sequence <math>\{\xi_i\}</math> converges in mean to <math>\xi</math> if
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| ::<math>\mbox{lim}_{i\rightarrow\infty}E[|\xi_i-\xi|]=0</math>.
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| '''Definition 4''': Suppose that <math>\Phi,\phi_1,\Phi_2,\ldots</math> are uncertainty distributions of uncertain variables <math>\xi,\xi_1,\xi_2,\ldots</math>, respectively. We say that the sequence <math>\{\xi_i\}</math> converges in distribution to <math>\xi</math> if <math>\Phi_i\rightarrow\Phi</math> at any continuity point of <math>\Phi</math>.
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| '''Theorem 1''': Convergence in Mean <math>\Rightarrow</math> Convergence in Measure <math>\Rightarrow</math> Convergence in Distribution.
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| However, Convergence in Mean <math>\nLeftrightarrow</math> Convergence Almost Surely <math>\nLeftrightarrow</math> Convergence in Distribution.
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| ==Conditional uncertainty==
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| '''Definition 1''': Let <math>(\Gamma,L,M)</math> be an uncertainty space, and <math>A,B\in L</math>. Then the conditional uncertain measure of A given B is defined by
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| ::<math>\mathcal{M}\{A\vert B\}=\begin{cases} \displaystyle\frac{\mathcal{M}\{A\cap B\} }{\mathcal{M}\{B\} }, &\displaystyle\text{if }\frac{\mathcal{M}\{A\cap B\} }{\mathcal{M}\{B\} }<0.5 \\ \displaystyle 1 - \frac{\mathcal{M}\{A^c\cap B\} }{\mathcal{M}\{B\} }, &\displaystyle\text{if } \frac{\mathcal{M}\{A^c\cap B\} }{\mathcal{M}\{B\} }<0.5 \\ 0.5, & \text{otherwise} \end{cases}</math>
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| ::<math>\text{provided that } \mathcal{M}\{B\}>0</math>
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| '''Theorem 1''': Let <math>(\Gamma,L,M)</math> be an uncertainty space, and B an event with <math>M\{B\} > 0</math>. Then M{·|B} defined by Definition 1 is an uncertain measure, and <math>(\Gamma,L,M\{\mbox{·}|B\})</math>is an uncertainty space.
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| '''Definition 2''': Let <math>\xi</math> be an uncertain variable on <math>(\Gamma,L,M)</math>. A conditional uncertain variable of <math>\xi</math> given B is a measurable function <math>\xi|_B</math> from the conditional uncertainty space <math>(\Gamma,L,M\{\mbox{·}|_B\})</math> to the set of real numbers such that
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| ::<math>\xi|_B(\gamma)=\xi(\gamma),\forall \gamma \in \Gamma</math>.
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| '''Definition 3''': The conditional uncertainty distribution <math>\Phi\rightarrow[0, 1]</math> of an uncertain variable <math>\xi</math> given B is defined by
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| ::<math>\Phi(x|B)=M\{\xi\leq x|B\}</math>
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| provided that <math>M\{B\}>0</math>.
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| '''Theorem 2''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi(x)</math>, and <math>t</math> a real number with <math>\Phi(t) < 1</math>. Then the conditional uncertainty distribution of <math>\xi</math> given <math>\xi> t</math> is
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| ::<math>\Phi(x\vert(t,+\infty))=\begin{cases} 0, & \text{if }\Phi(x)\le\Phi(t)\\ \displaystyle\frac{\Phi(x)}{1-\Phi(t)}\and 0.5, & \text{if }\Phi(t)<\Phi(x)\le(1+\Phi(t))/2 \\ \displaystyle\frac{\Phi(x)-\Phi(t)}{1-\Phi(t)}, & \text{if }(1+\Phi(t))/2\le\Phi(x) \end{cases}</math>
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| '''Theorem 3''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi(x)</math>, and <math>t</math> a real number with <math>\Phi(t)>0</math>. Then the conditional uncertainty distribution of <math>\xi</math> given <math>\xi\leq t</math> is
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| ::<math>\Phi(x\vert(-\infty,t])=\begin{cases} \displaystyle\frac{\Phi(x)}{\Phi(t)}, & \text{if }\Phi(x)\le\Phi(t)/2 \\ \displaystyle\frac{\Phi(x)+\Phi(t)-1}{\Phi(t)}\or 0.5, & \text{if }\Phi(t)/2\le\Phi(x)<\Phi(t) \\ 1, & \text{if }\Phi(t)\le\Phi(x) \end{cases}</math>
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| | |
| '''Definition 4''': Let <math>\xi</math> be an uncertain variable. Then the conditional expected value of <math>\xi</math> given B is defined by
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| ::<math>E[\xi|B]=\int_0^{+\infty}M\{\xi\geq r|B\}dr-\int_{-\infty}^0M\{\xi\leq r|B\}dr</math>
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| provided that at least one of the two integrals is finite.
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| ==References==
| |
| {{reflist}}
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| * Xin Gao, Some Properties of Continuous Uncertain Measure, ''[[International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems]]'', Vol.17, No.3, 419-426, 2009.
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| * Cuilian You, Some Convergence Theorems of Uncertain Sequences, ''Mathematical and Computer Modelling'', Vol.49, Nos.3-4, 482-487, 2009.
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| * Yuhan Liu, How to Generate Uncertain Measures, ''Proceedings of Tenth National Youth Conference on Information and Management Sciences'', August 3–7, 2008, Luoyang, pp. 23–26.
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| * Baoding Liu, Some Research Problems in Uncertainty Theory, ''Journal of Uncertain Systems'', Vol.3, No.1, 3-10, 2009.
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| * Yang Zuo, Xiaoyu Ji, Theoretical Foundation of Uncertain Dominance, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 827–832.
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| * Yuhan Liu and Minghu Ha, Expected Value of Function of Uncertain Variables, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 779–781.
| |
| * Zhongfeng Qin, On Lognormal Uncertain Variable, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 753–755.
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| * Jin Peng, Value at Risk and Tail Value at Risk in Uncertain Environment, ''Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China'', July 20–28, 2009, pp. 787–793.
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| * Yi Peng, U-Curve and U-Coefficient in Uncertain Environment, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 815–820.
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| * Wei Liu, Jiuping Xu, Some Properties on Expected Value Operator for Uncertain Variables, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 808–811.
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| * Xiaohu Yang, Moments and Tails Inequality within the Framework of Uncertainty Theory, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 812–814.
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| * Yuan Gao, Analysis of k-out-of-n System with Uncertain Lifetimes, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 794–797.
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| * Xin Gao, Shuzhen Sun, Variance Formula for Trapezoidal Uncertain Variables, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 853–855.
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| * Zixiong Peng, A Sufficient and Necessary Condition of Product Uncertain Null Set, ''Proceedings of the Eighth International Conference on Information and Management Sciences'', Kunming, China, July 20–28, 2009, pp. 798–801.
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| {{DEFAULTSORT:Uncertainty Theory}}
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| [[Category:Probability theory]]
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| [[Category:Fuzzy logic]]
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