PDF-PROBLEM5.LetX1;X2;:::beiidwithEXk=0andEX2k=2.ShowthatPnk=1Xk
Author : bitechmu | Published Date : 2020-11-19
16Pnk1X2k1712 indistributionwhere isthestandardnormaldistributionPROBLEM6SupposeX1X2arerandomvariablessuchthatthecentrallimittheoremp n22Xn0c 27 indistributionwhere isas
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PROBLEM5.LetX1;X2;:::beiidwithEXk=0andEX2k=2.ShowthatPnk=1Xk: Transcript
16Pnk1X2k1712 indistributionwhere isthestandardnormaldistributionPROBLEM6SupposeX1X2arerandomvariablessuchthatthecentrallimittheoremp n22Xn0c 27 indistributionwhere isas. b+c+1+b c+a+1+c a+b+1+(1 a)(1 b)(1 c)1:Problem4(USAMO1999).Leta1;a2;:::;an(n 3)berealnumberssuchthata1+a2++annanda21+a22++a2nn2:Provethatmax(a1;a2;:::;an)2.Problem5(USAMO1998).Leta1;:::;anbe page40110SOR201(2002)(ii)Bernoullir.v.{ifp1=p;p0=1 p=q;pk=0;k6=0or1,thenGX(s)=E(sX)=q+ps:(3:4)(iii)Geometricr.v.{ifpk=pqk 1;k=1;2;:::;q=1 p;thenGX(s)=ps1 qsifjsjq 1(seeHWSheet4.):(3:5)(iv)Binomialr.v. De nition Lemma LetCRnbeaconvexset.Ifx1;:::;xk2C,andzisaconvexcombinationofthexi,thenz2C. LeovanIersel(TUE) PolyhedraandPolytopes ORN42/22 De nition LetXRn.TheconvexhullofXisthesetofallconvexcombina 2REMCOVANDERHOFSTAD,NINAGANTERTANDWOLFGANGKONIGspace,letY=(Yz)z2Zdbeani.i.d.sequenceofrandomvariables,independentofthewalk.WerefertoYastherandomscenery.Thentheprocess(Zn)n2Nde nedbyZn=n 1Xk=0YSk;n2N; Problem5.35 Solution: FromEq.(5.2),1=B1 m1=1 m1( copy.jpg 2 (1+at)n+1;n=1;2;:::at 1+at;n=09 = ;(i)E[X(t)]=1X0nP(X(t)=n)=1X1n(at)n 1 (1+at)n+1=1 (1+at)21X1nat 1+atn 1=1 (1+at)2(1+at)2=1(ii)E[X2(t)]=Xn2(at)n 1 (1+at)n+1=1 (1+at)21X1n2at 1+atn 1=1 Problem5.25 r[1(4r+1)e4r] Solution: Wecanfolloweitheroftwopossibleapproaches.TherstinvolvestheuseofAmp Approach1:Amp ere'slaw ApplyingAmp ere'slawatr=anZCdjr=a=IZ2 r[1(4r+1)e4r] (~ )makinguseoftheno-tationin[14]where(~x)=Qdim~xk=1 (xk) (Pdim~xk=1xk)andwetreat~ asavectorofsizeKjwitheachvalueequalto .Notethatbecauseeachlabelhasitsowndistinctsubsetoftopics,thetopicassignmenta Problem5.27 (a) DetermineB (b) UseEq.(5.66)tocalculatethemagneticuxpassingthroughasquareloopwith0.25-m-longedgesiftheloopisinthex RGKing,BostonUniversityEC541Session24HouseholdsRepresentativehouseholdsolvesmaxE01Xt=0tU(Ct;Nt)(1)subjecttoPtCt+QtBtBt�1+WtNt+Dt(2)fort=0;1;2;:::plusasolvencyconstraint(tobediscussedfurth 1�pforjpj1;wewouldget1Xk=0kpk�1=1 (1�p)2;1 and1Xk=0k2pk�1=p (1�p)20=1+p (1�p)3forjpj1.Finally,wehave1Xk=0k2pk=p(1+p) q3:Plugginginthisexpression,itfollowsthata0=�1&# 1 2FIRSTTHINGSFIRST(5)Duringclasstheinstructorhasthenaldecisionondeterminingwhetheranar-gumentmaystandornot.Hisverdictmaystillbechallengedafteraproofis\published"(seerule(6)).(6)Ifsomeoneothertha 22n16XntLXnt17klItsl1normisupperboundedbykJk1maxpqXkljJklpqj122nmaxpqXkl1212121212300dntkl2XntLXntklXntpq12121212122022nmaxpqXkl12121212122XntLXntklXntpq121212121222n12121212122XntLXntklXntpq121212121 ToeplitzandCirculantMatrices:Areview RobertM.GrayDeptartmentofElectricalEngineeringStanfordUniversityStanford94305,USArmgray@stanford.edu Contents Chapter1Introduction11.1ToeplitzandCirculantMatrices1
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