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Krymsky Ufa State Aviation Technical University, Russia; E-mail: kvg@mail.rb.ruLEFH<   ;  XImprecise Prevision Theory (IPT) Starting Points: Fundamental Publications [1], [2] @Y ((4 (X [1] Walley P., Statistical reasoning with imprecise probabilities, Chapman and Hall, New York, (1991); [2] Kuznetsov V., Interval statistical models, Radio and Sviaz, Moscow, (1991) (in Russian).^P3* 0b  b   ,   8Traditional Problem Formulation in the Framework of IPT "97((7  Constraints:     &Dual for the Initial Problem Statement''(& PImportant Conclusion Concerning Optimal Solutions (L. Utkin and I. Kozine, [3])4Q2((H7     [3] Utkin L. and Kozine I. Different faces of the natural extension. In: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, ISIPTA '01, 2001, pp. 316-323.<)H     $Distribution of Probabilistic Masses%%($ / Masses are concentrated in the fixed points: 00/ Use of Additional Judgements("    Additional judgement can be reflected by inequality: r(x)K=const, (7) where is such that m6 &:  / Main Goal (Theorem)(  If there is no any finite interval for which function g(x) can be represented in the form (8) where then function (x) providing solution of optimization problem mentioned above, belongs to class of step-functions with minimum value equal to 0 and maximum value equal to K. gZZ>23""""{ l(          Some Comments(   To provide (8) the system  2Applying Methodology of the Calculus of Variations33(2  The inequalities (9) should be excluded from direct consideration in order to allow operating in the open domain with the values of the function. The requirement r(x)e"0 can be replaced by denoting:  # #Modified Formulation of the Problem$$(# / We would like to estimate and subject to B 0/ ) Lagrange Approach( / &The Necessary Conditions of Optimality''(& & The equations look here as follows:''& 9 Practical Implementation(  Optimal probability density: <&Reformulation of the Problem Statement''(&  We would like to estimate A Example 1 (   The information concerning a continuous random variable X is where K, T are fixed positive numbers. What are the bounds for the expectation M(X)? * * * Let us choose m=0. Objective function:92&  " F Solution of Optimization Problem!!(  L Lower and upper bounds of J interval:4M0L I Example 2 (   We add the constraint: where is the indicator function. Here also any finite interval of x values for which cannot be found, so the theorem can be applied. Further analysis shows, that m=1 is the best choice for such situation. TZu$d)@U f  G PExample 2 (Continuation 1)( ' To provide we have to set: (i) if((4     QExample 2 (Continuation 2)(  (ii) if    WAcknowledgements(  The research was initiated by Dr. Igor Kozine of Ris National Laboratory, Denmark, whose kind attention to this work is gratefully acknow-ledged. The work was partially supported by the grant T02-3.2-346 of Russian Ministry for Education which is also acknowledged. 4 ZZ b'    O     b` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴` fZXdbmo` ZXdbmo` ZXdbmo` 3fZXdbmo>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> ld(    6   V"1@075F 703>;>2:0 \  0 @  1@075F B5:AB0 B>@>9 C@>25=L "@5B89 C@>25=L '5B25@BK9 C@>25=L OBK9 C@>25=L M  0Ǚ }   X*  0̙ }    Z*  0љ }    Z*H  0޽h ? ZXdbmo80___PPT10.(H .D>@<;5=85 ?> C<>;G0=8N b Pd(     B[jJ "`j@   [ x  c $x[  [ 2  0["`u Kgs. 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GHYWձ{y9"X܏4r%NJLA> MaNC4X@k)'^xO)w9ow^ ~-UxXOA~3]PJ%ÆD'0$&F)@ED'YbiC{3jv y켷3|^OMjpds>)1n+~esJ.J0q`8Yp>'c9[JOo aW Z?=Ɍn0)^ecpf!6< [P 9c @T=ɘ(3-߫g02]q] HjQuV'? 蕗s"w2ĆPEa cƴvQV0?!ᴣ0qwʱ( (P4&=]Ee<>^x[d[0& V .x+).O>$k8X`%,g;n=R6!i$ gD(vvKUaEi jVmkgGLM#id [QєG14h(y't/U>"F9^2P Z;KQd̨sL~/kb"evzDޕ8靦̕L~csSC-HG⳰4LVqo+U2=51,6Kyy/*zex[ 2NlT)~}iB<6$(/<{ #c"HdQi)&]G*o8F#L_ %#9+K剼Mv[ /Jop7br`P|W5fhY pK$)M-pbi1s6:<>e@DG3 IJhLlO VƩAY\(`c[g6ilnpuy;{//5Əߗߝ{d<7 2ܳIJ KB /<7I[?  W Oh+'0 `h    $08XImprecise Prevision Theory (IPT) Starting Points: Fundamental Publications [1], [2] 175Microsoft PowerPointheo@E@ӍN(@%$OaGvg  3  - @ !--'-- $%(%(---'. @Times New Roman-$2 |Computing Interval (0  .Courier New((fb-.@Times New Roman--2 DEstimates for Components %/ '0 .-.@Times New Roman-02 Tof Statistical Information / .-.@Times New Roman-02 D4with Respect to Judgements '(/ .-.@Times New Roman-*2 uon Probability Density n #(.-.@Times New Roman-2   Functionsi#.--. @Arial-2 V Victor G.   " .-.@Arial-2 VKrymsky (.-.@Arial-32 Ufa State Aviation Technical           .-.@Arial-$2 University, Russia;T      .-.@Arial- 2 E.-.@Arial- 2 - .-.@Arial-2 mail: ! .-.@Arial-2 8kvg@mail.rb.ru&!  .----$yndnIo;o.p!qq rstuvxyz|}~yvspnmmmnpsvy~ !.;Id$/9BKT\ckqw}}wqkc\T}K|Bz9y/x$vutsrqqpoonn---%yndnIo;o.p!qq rstuvxyz|}~yvspnmmmnpsvy~ !.;Id$/9BKT\ckqw}}wqkc\T}K|Bz9y/x$vutsrqqpoonn---'.@Arial- 2 Kgs.-.@Arial- 2 . .-.@Arial-2 Lyngby.-.@Arial-2 d, Denmark, 2004   .- ՜.+,0    e9A ArialTimes New RomanSymbol Microsoft Equation 3.0Computing Interval Estimates for Components of Statistical Information with Respect to Judgements on Probability Density FunctionsYImprecise Prevision Theory (IPT) Starting Points: Fundamental Publications [1], [2] 9Traditional Problem Formulation in the Framework of IPT 'Dual for the Initial Problem StatementQImportant Conclusion Concerning Optimal Solutions (L. Utkin and I. Kozine, [3])%Distribution of Probabilistic MassesUse of Additional JudgementsMain Goal (Theorem)Some Comments3Applying Methodology of the Calculus of Variations$Modified Formulation of the ProblemLagrange Approach'The Necessary Conditions of OptimalityPractical Implementation'Reformulation of the Problem Statement Example 1!Solution of Optimization Problem Example 2Example 2 (Continuation 1)Example 2 (Continuation 2)Acknowledgements    OLE _qu  !"#$%&'()*+,-./012345789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%'()*+,-/012345:Root EntrydO)PictureskCurrent User.SummaryInformation(PowerPoint Document(6DocumentSummaryInformation8&