ࡱ> \^[z5@ bjbj22 -XXP@n    $ ...8@..d$ |d/x4"444==D>$ """"""$eR:F] C ;<"CCF 44ZoMoMoMC| 84 4 oMC oMjoMN,  l4X/ 0EȢO.'J$0f KT,l$ $  l>>:x?oMX@ A>>>>>>FF$ $ !.GM($ $ .Module PE.PAS.U8.3 Bivariate Random Variables Primary Author: James D. McCalley, Iowa State University Email Address: jdm@iastate.edu Co-author: None Email Address: None Last Update: 8/1/02 Reviews: None Prerequisite Competencies: 1. Know basic probability relations (module U5) 2. Define random variables and distinguish between discrete (module U6) and continuous (module U7) random variables. 3. Relate probability density functions and distributions for discrete and continuous random variables (module U7). Module Objectives: 1. Use and relate joint, marginal, and conditional pdfs and cdfs. U8.1 Introduction I n this module, we describe the so-called bivariate case of uncertain situations where two quantities are modeled as random variables. Associated analytic models for density functions and distributions, and the relations between then, will be discussed. Although the bivariate case occurs in many disciplines, we will confine our discussion to examples associated with power system engineering. It is important to note that the bivariate case is the simplest of the more general multivariate case and as such, serves well to illustrate some fundamental concepts associated with uncertain situations requiring multivariate random variables. This module addresses only the continuous bivariate case; similar treatment may be accorded the discrete bivariate case. U8.2 A Motivating Illustration Power system engineers have always computed current ratings for transmission conductors. This is necessary because excessive currents lead to conductor heating, expansion of the conductor material, sagging, and violation of clearance requirements (as set by law). The approach is to define a maximum conductor temperature, (max, beyond which it is unsafe to operate, and then compute the conductor temperature under various currents i until the current is found for which the conductor temperate exceeds (max. What one must recognize is, however, that the conductor temperature depends on other factors besides current alone, the most significant of which are current duration d, ambient temperature t (the temperature of the air surrounding the conductor), and normal wind speed u (the component of the wind velocity that is perpendicular to the conductor). There are other less influential factors as well, including solar radiation, but we will ignore these for the moment. Thus, we see that conductor temperature ( may be written as a function of i, d, t, and u. Lets denote the function characterizing this relation as g, so that  EMBED Equation.3  (U8.1) We simplify the problem by making the following assumptions: The current is fixed at a high level of i0, and we desire to see if the conductor temperature is within its safe range for a time period between 3 and 4 pm on a summer day. The current level of i0 will last for a duration of d=1hour. The transmission line is very short so that the temperature and wind speed conditions have no spatial variation, i.e., these conditions are the same at one point on the line as another. The ambient temperature and wind speed both vary during the next hour, but their variation is independent. Thus, we may treat them as independent random variables T and U, respectively. We have information that allows us to characterize the uncertainty associated with the ambient temperature t and the wind speed u during this hour, according to the following uniform probability density functions (pdfs).  EMBED Equation.3   EMBED Equation.3  (U8.2) We are able to utilize (U8.1) to identify the range of safe operating conditions according to the following relation:  EMBED Equation.3  (U8.3) The inequality of (U8.3) for safe conductor operation can be illustrated on a diagram of the possible range and possible safe range of t and u, as indicated in Fig. U8.1.  EMBED Word.Picture.8  Fig. U8.1: Illustration of possible and safe ranges of conductor operation We desire to compute the probability that the conductor will be operating safely in the hour of interest. This is the probability that the conductor temperature does not exceed (max, i.e., Pr((<(max). It is also the probability that the (T,U) random variable combination falls within the safe range of operation, which can be expressed as Pr(201, y>1. 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New York, 1999. L. Bain and M. Engelhardt, Introduction to Probability and Mathematical Statistics, Duxbury Press, Belmont, California, 1992. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 1984. W. Blischke and D. Murthy, Reliability: Modeling, Prediction, and Optimization, John Wiley, New York, 2000. R. Billinton and R. Allan, Reliability Evaluation of Engineering Systems, Plenum Press, New York, 1992. PAGE  PAGE 11 Module U8: Bivariate Random Variables  INCLUDEPICTURE "http://powerlearn.ee.iastate.edu/test/test3_files/PLLogoSmall.gif" \* MERGEFORMATINET  L̜1  GHI!h]h&`#$* !GIJjhxUjhxUhx hxCJjhxCJUmHnHu +0&P1hP/ =!"#$%@Dd @J  C A? "27 g^vZ`c.~D `!v7 g^vZ`c.@ Dxcdd``Ndd``baV d,FYzP1n:&B@?b u p00 UXRY7S?&,e`abM-VK-WMcZ>  X.4+Q$Y au"T~33QF? +nO]%p;R|F0?_Wb '`3W&0;p]pw}fͿL,hrHs]#ֿ \.] `pbdbR ,.I@X`lTmDd ` `J  C A? 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F T,U (t,u)=f T,U (,)d ""u +" ""t +" d=0.02d 5u +" 20t +" d=0.02ut"0.4u"0.1t+2Equation Native w_1092121169kF*¢O*¢OOle CompObjjlf FMicrosoft Equation 3.0 DS Equation Equation.39qH<  F T,U (t,u)=0.02ut"0.4u"0.1t+2,20d"td"30,and5d"ud"100ObjInfomEquation Native _1092057917dspF*¢O*¢OOle ,t<20oru<50.1t"2,20d"td"30andu>100.2u"1,t>30and5d"ud"10{ FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjoqfObjInforEquation Native _1092058061uF*¢O*¢OMc_$ Pr({T,U}"safe range)=Pr(20<T<30, 0.5T"5<U<10)                                       =f T,U (t,u)dudt saferange +"+" =0.02dudt 0.5t"510 +" 2030 +" FMicrosoft Equation 3.0 DS Equation Equation.39qM@ Pr[(ad"Xd"b))"(cd"Yd"d)]=fOle CompObjtvfObjInfowEquation Native  X,Y (,)dd cd +" ab +" FMicrosoft Equation 3.0 DS Equation Equation.39qM$ Pr[(ad"Xd"b))"(""d"Yd"")]=_1092058892nzF*¢O*¢OOle CompObjy{fObjInfo|Equation Native _1092061336xF*¢O*¢OOle CompObj~ff X,Y (x,y)dyd """ +" ab +" x FMicrosoft Equation 3.0 DS Equation Equation.39qM~Мԩ f X (x)=f X,Y (x,y)dyObjInfoEquation Native _1092058970F*¢O*¢OOle  """ +" FMicrosoft Equation 3.0 DS Equation Equation.39qM~P f Y (y)=f X,Y (x,y)dx """ +"CompObjfObjInfoEquation Native _1092060064F*¢O*¢OOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qMPT f U (u)=f T,U (t,u)dt= """ +" 0.02dt= 2030 +" 0.25d"ud"10 FMicrosoft Equation 3.0 DS Equation Equation.39qMx_ f T (t)=f T,U (t,u)du= """ +" 0.02du= _1092060198F*¢O*¢OOle CompObjfObjInfoEquation Native _1092060927F*¢O*¢OOle CompObjf510 +" 0.120d"td"30 FMicrosoft Equation 3.0 DS Equation Equation.39qMnt F X (x)=f X ()d ""xObjInfoEquation Native _1092061911F*¢O*¢OOle  +" FMicrosoft Equation 3.0 DS Equation Equation.39qM¤  F X (x)=f X,Y (,y)dy """ +" d ""x +"CompObjfObjInfoEquation Native _1092061444F*¢O*¢OOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qM¤`$ F Y (y)=f X,Y (x,)dx """ +" d ""y +"_1092062479F*¢O*¢OOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qMNh F T (t)=f T,U (,u)du """ +" d ""t +" =0.02du 510 +" d 20t +" =0.1t"2,Equation Native j_1092062413F*¢O*¢OOle CompObjf20d"td"30 FMicrosoft Equation 3.0 DS Equation Equation.39qMJd F U (u)=f T,U (t,)dt """ +" d ""u +" ObjInfoEquation Native f_1092112416F*¢OPâOOle   #$%(+,-0347:;<=>ADEFGHIJMPQRSTWZ[\]^adefghilopqtwxyz{~=0.02dt 2030 +" d 5u +" =0.2u"1,5d"ud"10 FMicrosoft Equation 3.0 DS Equation Equation.39qMbL Pr(A|B)CompObjfObjInfoEquation Native ~_1092112685FPâOPâO<;      !"#$%&'()*+,-./0123456789:=?>@ABCDEFHGIJLKMNOPQRSTUVWX[Y\{]^_`bacdefghijklmnopqrstvuxwzy|}~=Pr(A)"B)Pr(B) FMicrosoft Equation 3.0 DS Equation Equation.39qM- A:  [Xd"x]Ole  CompObj fObjInfo Equation Native  I7Fɠ?HP"GP_|Ѓϓ~ӿ݇~n+׊nVt+ ܞ:ځt@ O*\[m@u++4n\.IMA:{ N$/_p  8e1p2B543F^\@jzhƀ|A/7>)_)M=C⺕f76S1MR9XV ZNo( 6ҫz6.oD q煴^4BhѪ%2Z2 `<GQi7Ec'ʥC4 :aazJ1@z?DxS3 zPS(;1r6܉z3گF][1o-nگ3Fqg+#_k3 h{1+K^ 32zS<75ϏY(M#%x8-헁Y'p=|Rb'1hy?8~8Nd*;-\'׸~#&By;4}1?C1[l%@`-X'^orXp]*VcE"-KT'M..u ;<}=BYY) 'E(!i&_A>* 5]t!s ɣhK^%z!?Ɯمs/V΃`ū?LaoL /g8su^ys8w TWb#vI^~PRŞ+k>b 6h, ǯ[fl9s9vcW9v V\>kKx)vʷ(3c|e_:œR/!-yfG_#Ը WK'k,/wVfUɍ3>Q\A5u*?lv[P }dBBKnx՚Xeh,m_t ?^^6/0e!,mA ?`iNN)(|;qe~GUG٠r I3t~ A>DfAe~{ۧlHl=}lP nN;IlFG`$m}l9)ޑ%#CXGȟ~ۡTQ6yIt*(}6o+t*(lySg$SI'p/" y":~QWX &Bv2A} &\lyNh7] k*njŻ.kÞE}O"_e=uBr$r~6{9GpA.䧕7 90\`(ӧ{Y$J_8M4Uj_ Aȩw5d:~O ?H3,~'Hڴ_:w<Nb7iyG0{@NUz;cxx @c112BYL%bpuulREf0 Dd 4dJ  C A? 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KA?H1: ㆪaM,,He` @201d++&_1092112725FPâOPâOOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qMA8` B:  [yd"Yd"y+y] FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native ]_1092112826}FPâOPâOOle CompObjfObjInfoEquation Native  _1092113637FPâOPâOOle M Pr(A|B)=Pr(A)"B)Pr(B)=Pr[(Xd"x))"(yd"Yd"y+y)]Pr(yd"Yd"y+y) FMicrosoft Equation 3.0 DS EqCompObjfObjInfo!Equation Native "_1092113811FPâOPâOuation Equation.39qMД F X (x|(yd"Yd"y+y))=Pr[(Xd"x))"(yd"Yd"y+y)]Pr(yd"Yd"y+y) FMicrosoft Equation 3.0 DS EqOle &CompObj'fObjInfo)Equation Native *uation Equation.39qM Pr[(Xd"x))"(yd"Yd"y+y)]=f X,Y (,)dd yy+y +" ""x +" FMicrosoft Equation 3.0 DS Eq_1092113964FPâOPâOOle .CompObj/fObjInfo1uation Equation.39qMˆ@ Pr(yd"Yd"y+y)=f Y ()d yy+y +" FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native 2_1092113912FPâOPâOOle 5CompObj6fObjInfo8Equation Native 9H_1092114842FPâOPâOOle ?M,4 F X (x|(yd"Yd"y+y))=f X,Y (,)dd yy+y +" ""x +" f Y ()d yy+y +"CompObj@fObjInfoBEquation Native C_1092115025FPâOPâO FMicrosoft Equation 3.0 DS Equation Equation.39qM¦@ F X (x|y)='lim y!0 F X (x|(yd"Yd"y+y))='lim y!0 f X,Y (,)dd yy+y +" ""x +" f Y ()d yy+y +" FMicrosoft Equation 3.0 DS Equation Equation.39qMXL F X (x|Ole KCompObjLfObjInfoNEquation Native Oty)='lim y!0 f X,Y (,)dd yy+y +" ""x +" 'lim y!0 f Y ()d yy+y +" FMicrosoft Equation 3.0 DS Equation Equation.39q_1092115701FPâOPâOOle UCompObjVfObjInfoXMXt F X (x|y)='lim y!0 f X,Y (,)dd yy+y +" ""x +" 'lim y!0 f Y ()d yy+y +"Equation Native Yt_1092115813FPâOPâOOle _CompObj`f FMicrosoft Equation 3.0 DS Equation Equation.39qMr 'lim y!0 f X,Y (,)dd yy+y +" ""x +" =f X,Y (,y) ""x +" yd=yfObjInfobEquation Native c_1092115869FPâOPâOOle j X,Y (,y) ""x +" d FMicrosoft Equation 3.0 DS Equation Equation.39qM§bl 'lim y!0 f Y ()d yCompObjkfObjInfomEquation Native n_1092125490FPâOPâOy+y +" =yf Y (y) FMicrosoft Equation 3.0 DS Equation Equation.39q=xK  F X (x|y)=yf X,Y (Ole rCompObjsfObjInfouEquation Native vY,y)d ""x +" yf Y (y)=f X,Y (,y)d ""x +" f Y (y) FMicrosoft Equation 3.0 DS Equation Equation.39q_1092116482FPâOPâOOle |CompObj}fObjInfoEquation Native _1092125061FPâO7ŢOOle CompObjfM® F Y (y|x)=f X,Y (x,)d ""y +" f X (x) FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1092125079i F7ŢO7ŢOOle oG  f X ()d ""x +" =F X (x) FMicrosoft Equation 3.0 DS Equation Equation.39q^8 f X (x)CompObjfObjInfoEquation Native z_1092125284F7ŢO7ŢO=dF X (x)dx FMicrosoft Equation 3.0 DS Equation Equation.39qW f X (|y)d ""x +" =F X (x|y)Ole CompObjfObjInfoEquation Native _1092125369F7ŢO7ŢOOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qn`4 f X (x|y)=dF X (x|y)dxEquation Native _1092125506 F7ŢO7ŢOOle CompObj  f FMicrosoft Equation 3.0 DS Equation Equation.39qjи, f X (x|y)=dF X (x|y)dx=ddxf X,Y (,y)d ""x +" f Y (y){}=f ObjInfo Equation Native _1092125646F7ŢO7ŢOOle X,Y (x,y)f Y (y) FMicrosoft Equation 3.0 DS Equation Equation.39qjE f X (y|x)=dF Y (y|x)CompObjfObjInfoEquation Native _1092124542F7ŢO7ŢOdy=ddyf X,Y (x,)d ""x +" f X (x){}=f X,Y (x,y)f X (x) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native I  F T (t|u)=f T,U (,y)d ""t +" f U (u)=0.02d 20t +" 0.2=0.02t"0.40.2=0.1t"2,20d"td"30_1092124826F7ŢO7ŢOOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q{O F U (u|t)=f T,U (,y)d ""u +" f T (t)=0.02d 5u +Equation Native _1092126785"F7ŢO7ŢOOle CompObj f" 0.1=0.02u"0.10.1=0.2u"1,5d"td"10 FMicrosoft Equation 3.0 DS Equation Equation.39q f X,Y (ObjInfo!Equation Native _1092127837,'$F7ŢO7ŢOOle x,y)=x 2 +xy3,0d"xd"1,0d"yd"2 FMicrosoft Equation 3.0 DS Equation Equation.39qxl f X (x)CompObj#%fObjInfo&Equation Native _109212809416)F7ŢO7ŢO=f X,Y (x,y)dy=x 2 +xy3dy=x 2 y+ 02 +" """ +" xy 2 6 02 =2x 2 +23x,0d"xd"10otherwise{Ole CompObj(*fObjInfo+Equation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q,o f Y (y)=f X,Y (x,y)dx=x 2 +xy3dx=x 3 3+ 01 +" """ +" x 2 y6 01 =13+16y,0d"yd"20otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39q f X (x|_1092127679.F7ŢO7ŢOOle CompObj-/fObjInfo0Equation Native _10921278463F7ŢO7ŢOOle CompObj24fy)=f X,Y (x,y)f Y (y)=x 2 +xy313+16y=6x 2 +2xy2+y0d"xd"1,0d"yd"20otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39q f X (y|x)=f X,Y (x,y)f X (x)=x 2 +xy32x 2 +23x=3x+y6x+ObjInfo5Equation Native _10921281868F7ŢOƢOOle      !$'()*+,/234569<=>?@CFGHIJMPQRSUVWXYZ\]^_a20d"xd"1,0d"yd"20otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39q`L f X (x)f Y (y)=(2x 2 +CompObj79fObjInfo:Equation Native _1006083399=FƢOƢO23x)(13+16y)=23x 2 +13x 2 y+29x+19y`"f X,Y (x,y)=x 2 +xy3 FMicrosoft Equation 3.0 DS Equation Equation.39qOle  CompObj<>fObjInfo?Equation Native lba=P$vIpI f X,Y (x,y)=xy,    0d"xd"1,    0d"yd"20,                              otherwise{_1006084806;EBFƢOƢOOle CompObjACfObjInfoD FMicrosoft Equation 3.0 DS Equation Equation.39q=ؔ$vIpI f X,Y (x,y)=c(x 1/3 +y 1/5 ),    0d"xd"60,    0d"yd"100,              Equation Native _1008422294YGFƢOƢOOle "CompObjFH#f                otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39qҐ$vIpI f X,Y (x,y)=9"e "3ObjInfoI%Equation Native &_1008422903@LFƢOƢOOle -x "e "3y ,                x,ye"00,                              otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjKM.fObjInfoN0Equation Native 1_1008423084TQFƢOƢOd$vIpI f X,Y (x,y)=4225"xy,    0d"xd"3,  0d"yd"50,                              otherwise{Ole 7CompObjPR8fObjInfoS:Equation Native ; FMicrosoft Equation 3.0 DS Equation Equation.39qd$vIpI f X,Y (x,y)=4225"xy,    0d"xd"3,  0d"yd"50,                              otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39qd$vIpI f X,Y (x,y)=A"x+y(),    0d"xd"1,  0_1008423256VFƢOƢOOle ACompObjUWBfObjInfoXDEquation Native E_1006245005[FƢOƢOOle KCompObjZ\Lfd"yd"10,                              otherwise{ FMicrosoft Equation 3.0 DS Equation Equation.39q=kIvI f X,Y (ObjInfo]NEquation Native O,1TableSummaryInformation(`Tx,y)=4xy,   0d"xd"1,   0d"yd"10,               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