Intro-Probability_2nd_Problem_Solution

《Intro-Probability_2nd_Problem_Solution》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、IntroductiontoProbability2ndEditionProblemSolutions(lastupdated:7/31/08)cDimitriP.BertsekasandJohnN.TsitsiklisMassachusettsInstituteofTechnologyWWWsiteforbookinformationandordershttp://www.athenasc.comAthenaScientific,Belmont,Massachusetts1CHAPTER1SolutiontoProblem1.1.WehaveA={2,4,6},B={4,5,6}

2、,soA∪B={2,4,5,6},andc(A∪B)={1,3}.Ontheotherhand,ccA∩B={1,3,5}∩{1,2,3}={1,3}.Similarly,wehaveA∩B={4,6},andc(A∩B)={1,2,3,5}.Ontheotherhand,ccA∪B={1,3,5}∪{1,2,3}={1,2,3,5}.SolutiontoProblem1.2.(a)ByusingaVenndiagramitcanbeseenthatforanysetsSandT,wehavecS=(S∩T)∪(S∩T).c(Alternatively,arguethatanyx

3、mustbelongtoeitherTortoT,soxbelongstoSccifandonlyifitbelongstoS∩TortoS∩T.)ApplythisequalitywithS=AandT=B,toobtainthefirstrelationccccA=(A∩B)∪(A∩B).InterchangetherolesofAandBtoobtainthesecondrelation.(b)ByDeMorgan’slaw,wehaveccc(A∩B)=A∪B,andbyusingtheequalitiesofpart(a),weobtain

ccccccccccc(

4、A∩B)=(A∩B)∪(A∩B)∪(A∩B)∪(A∩B)=(A∩B)∪(A∩B)∪(A∩B).(c)WehaveA={1,3,5}andB={1,2,3},soA∩B={1,3}.Therefore,c(A∩B)={2,4,5,6},2andccccA∩B={2},A∩B={4,6},A∩B={5}.Thus,theequalityofpart(b)isverified.SolutiontoProblem1.5.LetGandCbetheeventsthatthechosenstudentisageniusandachocolatelover,respectively.Wehave

5、P(G)=0.6,P(C)=0.7,andccP(G∩C)=0.4.WeareinterestedinP(G∩C),whichisobtainedwiththefollowingcalculation:
ccP(G∩C)=1−P(G∪C)=1−P(G)+P(C)−P(G∩C)=1−(0.6+0.7−0.4)=0.1.SolutiontoProblem1.6.Wefirstdeterminetheprobabilitiesofthesixpossibleoutcomes.Leta=P({1})=P({3})=P({5})andb=P({2})=P({4})=P({6}).Weare

6、giventhatb=2a.Bytheadditivityandnormalizationaxioms,1=3a+3b=3a+6a=9a.Thus,a=1/9,b=2/9,andP({1,2,3})=4/9.SolutiontoProblem1.7.Theoutcomeofthisexperimentcanbeanyfinitesequenceoftheform(a1,a2,...,an),wherenisanarbitrarypositiveinteger,a1,a2,...,an−1belongto{1,3},andanbelongsto{2,4}.Inaddition,the

7、rearepossibleoutcomesinwhichanevennumberisneverobtained.Suchoutcomesareinfinitesequences(a1,a2,...),witheachelementinthesequencebelongingto{1,3}.Thesamplespaceconsistsofallpossibleoutcomesoftheabovetwotypes.SolutiontoProblem1.8.Letpibetheproba