第三届丘成桐大学生数学竞赛试题-概率统计及应用数学.docxVIP

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INDIVIDUALTEST

S.-TYAUCOLLEGEMATHCONTESTS2012

ProbabilityandStatistics

Pleasesolve5outofthefollowing6problems,orhighestscoresof5problemswillbecounted.

1.Solvethefollowingtwoproblems:

1)Anurncontainsbblackballsandrredballs.Oneoftheballswasdrawnatrandom,andputtedbackintheurnwithaadditionalballsofthesamecolor.Nowsupposethatthesecondballdrawnatrandomisred.Whatistheprobabilitythatthefirstballdrawnwasblack?

2)Let(Xn)beasequenceofrandomvariablessatisfying

Assumethatsequenceofrandomvariables(Yn)convergesto0inprob-ability.Provethat(XnYn)convergesto0inprobability

2.Solvethefollowingtwoproblems:

1)Let(Ω,F,P)beaprobabilityspace,Gbeasub-algebraofF.AssumethatXisanon-negativeintegrablerandomvariable.SetY=E[X|9].Provethat

(a)[X0]c[Y0],a.s.;

(b)[Y0j=ess.inf{A:A∈G,[X0]cA}.

2)LetXandYhaveabivariatenormaldistributionwithzeromeans,varianceso2andr2,respectively,andcorrelationp.Findthecondi-tionalexpectationE(X|X+Y).

3.Supposethat{p(i,j):i=1,2,·…,m;j=1,2,…,n}isafinite

bivariatejointprobabilitydistribution,thatis,

(i)Can{p(i,j)}bealwaysexpressedas

forsomefiniteAk≥0,∑kAk=1,ak(i)≥0,∑i=1ak(i)=1,bk(j)≥

0,

2

(ii)Proveordisprovetheaboverelationbyuseofconditionalprob-ability.

4.LetX?,…,Xmbeanindependentandidenticallydistributed(i.i.d.)randomsamplefromacumulativedistributionfunction(CDF)F,andY1,.,Ynani.i.d.randomsamplefromaCDFG.WewanttotestH?:F=GversusH?:F≠G.ThetotalsamplesizeisN=m+n.Considerthefollowingtwononparametrictests.

●TheWilcoxonranksumtests.Thetestproceedsbyfirstrank-ingthepooledXandYsamplesandthentakingthesumoftheranksassociatedwiththeYsample.LetRy,…,Rybether