2016年电大概率统计试题及答案(本科完整版)小抄参考

一、填空题（每题2分，共20分）1、记三事件为A，B，C则用A，B，C及其运算关系可将事件，“A，B，C中只有一个发生”表示为2、匣中有2个白球，3个红球。现一个接一个地从中随机地取出所有的球。那么，白球比红球早出现的概率是2/5。3、已知PA03，P（B）05，当A，B相互独立时，。06505P_,|_4、一袋中有9个红球1个白球，现有10名同学依次从袋中摸出一球（不放回），则第6位同学摸出白球的概率为1/10。5、若随机变量在区间上服从均匀分布，则对以及任意的正数，X,ABACB0E必有概率PCXE,CEB,BA6、设服从正态分布，则N32,42X2,N23XY7、设11863BEDN,PN_P_（且，则8、袋中装有5只球，编号为1，2，3，4，5，在袋中同时取出3只，以表示取出3只X球中的最大号码。则的数学期望45。9、设随机变量的分布律为,XYXY12310120100282018001230015005则条件概率2/5|YXP10、设来自正态总体,当常12,10N219285241IIIIIIXXY数1/4时，服从分布。KKY2二、计算题（每小题10分，共70分）1、三台机器因故障要人看管的概率分别为01，02，015，求（1）没有一台机器要看管的概率（2）至少有一台机器不要看管的概率（3）至多一台机器要看管的概率解以AJ表示“第J台机器需要人看管”，J1，2，3，则PA101,PA202,PA3015,由各台机器间的相互独立性可得312309850612121197313232312112308509850950985664PAAPA2、甲袋中有N只白球、M只红球；乙袋中有N只白球、M只红球。今从甲袋任取一球放入乙袋后，再从乙袋任取一球。问此球为白球的概率是多少解以W甲表示“第一次从甲袋取出的为白球”，R甲表示“第一次从甲袋取出的为红球”，W乙表示“第二次从乙袋取出的为白球”，则所求概率为PWPWR乙乙111NMNNMMNMCCN3、设随机变量X的概率密度为,试求（1）常数ACOS,|20AXF乙2分布函数3概率。FX4PX解（1）由归一性可得，从而212FXDACOSXD12A222XXXF,FFDDXF,0212,XSIN,40123024PXCOSXD4、（1）已知X的分布律为10123P613计算。（5分）2D解222414XEX15235464（2）、设，求的概率密度（5分）,0N2Y解Y的密度函数为210YE,FY5、设的概率密度为,X0,XYEF乙1试求分布函数；,YXF2求概率其中区域由轴,轴以及直线所围成PGXY1YX解001XYXYED,F,D,乙1XY,X,Y,乙2GPX,YFX,YD102XYEDE6、设二维随机变量的概率密度为,求常数及,XY1,0,KXF乙K边缘概率密度并讨论随机变量的相互独立性。,解由归一性知011,YXFXDKDY016K6,XFXFYD001XDYX乙601X乙,YFYFX16Y乙230，其他YY显然，故X与Y不相互独立。,XYF7、设总体的概率密度为,其中为未知参数若1,0XF乙0是来自母体的简单子样，试求的矩估计与极大似然估计NX,1解（1）令110EXXD解得的矩估计为2X（2）似然函数1121NNIIILXX对数似然函数1LLL2NII令1LNL0NIILX解得的极大似然估计为21LNIIX三、证明题（每题5分，共10分）1、为来自总体X的样本，证明当时，为总体均值的2,XAB12AXBEX无偏估计。证明设总体均值，由于为来自总体X的样本，E12,因此12而为总体均值的无偏估计，故应该有2AB112XABAB从而2、设是相互独立的随机变量，它们分别服从参数为的泊松分布，证明,XY21,服从参数为的泊松分布。Z21证明由题知，即12,XPY12,MNPEPE令，且由的相互独立性可得XY,ZKK0,MPIYKI120IKIKIE12120IKIIE1212,即服从参数为的泊松分布XY1WINGERTUIVASASHECK,WHOSCOREDTWOTRIESINTHEKIWIS2018SEMIFINALWINOVERENGLAND,HASBEENPASSEDFITAFTERALOWERLEGINJURY,WHILESLATERHASBEENNAMEDATFULLBACKBUTISSTILLRECOVERINGFROMAKNEEINJURYAGGRAVATEDAGAINSTUSABOTHSIDESBOAST100RECORDSHEADINGINTOTHEENCOUNTERBUTAUSTRALIAHAVENOTCONCEDEDATRYSINCEJOSHCHARNLEYSEFFORTINTHEIRFIRSTPOOLMATCHAGAINSTENGLANDONTHEOPENINGDAYAUSSIEWINGERJARRYDHAYNEISTHECOMPETITIONSTOPTRYSCORERWITHNINE,CLOSELYFOLLOWEDBYTUIVASASHECKWITHEIGHTBUTITISRECENTLYNAMEDRUGBYLEAGUEINTERNATIONALFEDERATIONPLAYEROFTHEYEARSONNYBILLWILLIAMSWHOHASATTRACTEDTHEMOSTINTERESTINTHETOURNAMENTSOFARTHEKIWIWITHATOURNAMENTHIGH17OFFLOADSHASTHECHANCEOFBECOMINGTHEFIRSTPLAYERTOWINTHEWORLDCUPINBOTHRUGBYLEAGUEANDRUGBYUNIONAFTERTRIUMPHINGWITHTHEALLBLACKSIN2011“IDGIVEEVERYAWARDBACKINAHEARTBEATJUSTTOGETACROSSTHELINETHISWEEKEND,“SAIDWILLIAMSTHELACKOFAIRUPTHEREWATCHMCAYMANISLANDSBASEDWEBB,THEHEADOFFIFASANTIRACISMTASKFORCE,ISINLONDONFORTHEFOOTBALLASSOCIATIONS150THANNIVERSARYCELEBRATIONSANDWILLATTENDCITYSPREMIERLEAGUEMATCHATCHELSEAONSUNDAY“IAMGOINGTOBEATTHEMATCHTOMORROWANDIHAVEASKEDTOMEETYAYATOURE,“HETOLDBBCSPORT“FORMEITSABOUTHOWHEFELTANDIWOULDLIKETOSPEAKTOHIMFIRSTTOFINDOUTWHATHISEXPERIENCEWAS“UEFAHASOPENEDDISCIPLINARYPROCEEDINGSAGAINSTCSKAFORTHE“RACISTBEHAVIOUROFTHEIRFANS“DURINGCITYS21WINMICHELPLATINI,PRESIDENTOFEUROPEANFOOTBALLSGOVERNINGBODY,HASALSOORDEREDANIMMEDIATEINVESTIGATIONINTOTHEREFEREESACTIONSCSKASAIDTHEYWERE“SURPRISEDANDDISAPPOINTED“BYTOURESCOMPLAINTINASTATEMENTTHERUSSIANSIDEADDED“WEFOUNDNORACISTINSULTSFROMFANSOFCSKA“BAUMGARTNERTHEDISAPPOINTINGNEWSMISSIONABORTEDTHESUPERSONICDESCENTCOULDHAPPENASEARLYASSUNDATHEWEATHERPLAYSANIMPORTANTROLEINTHISMISSIONSTARTINGATTHEGROUND,CONDITIONSHAVETOBEVERYCALMWINDSLESSTHAN2MPH,WITHNOPRECIPITATIONORHUMIDITYANDLIMITEDCLOUDCOVERTHEBALLOON,WITHCAPSULEATTACHED,WILLMOVETHROUGHTHELOWERLEVELOFTHEATMOSPHERETHETROPOSPHEREWHEREOURDAYTODAYWEATHERLIVESITWILLCLIMBHIGHERTHANTHETIPOFMOUNTEVEREST55MILES/885KILOMETERS,DRIFTINGEVENHIGHERTHANTHECRUISINGALTITUDEOFCOMMERCIALAIRLINERS56MILES/917KILOMETERSANDINTOTHESTRATOSPHEREASHECROSSESTHEBOUNDARYLAYERCALLEDTHETROPOPAUSE,ECANEXPECTALOTOFTURBULENCETHEBALLOONWILLSLOWLYDRIFTTOTHEEDGEOFSPACEAT120,000FEETTHEN,IWOULDASSUME,HEWILLSLOWLYSTEPOUTONTOSOMETHINGRESEMBLINGANOLYMPICDIVINGPLATFORMTHEYBLEWITIN2008WHENTHEYGOTCAUGHTCOLDINTHEFINALANDTHEYWILLNOTMAKETHESAMEMISTAKEAGAINSTTHEKIWISINMANCHESTERFIVEYEARSAGOTHEYCRUISEDTHROUGHTOTHEFINALANDSOFARHISTORYHASREPEATEDITSELFHERETHELASTTRYTHEYCONCEDEDWASSCOREDBYENGLANDSJOSHCHARNLEYINTHEOPENINGGAMEOFTHETOURNAMENTTHATCOULDBECLASSEDASAWEAKNESS,ATEAMUNDERCOOKEDBUTIHAVEBEENIMPRESSEDBYTHEKANGAROOSFOCUSINTHEIRGAMESSINCETHENTHEYHAVEBEENCONCENTRATINGONTHESORTOFSTUFFTHATWINSYOUTOUGH,EVENCONTESTSSTRONGDEFENCE,ESPECIALLYONTHEIROWNGOALLINE,COMPLETINGSETSANDAGOODKICKCHASETHEYVEBEENGREATATALLTHEUNGLAMOROUSSTUFFTHATOFTENGOESUNNOTICEDINTHESTANDSBUTNOTBYYOURTEAMMATESITISASTHOUGHTHEIRENTIRETOURNAMENTHASBEENPREPARATIONFORTHEFINALINJOHNATHANTHURSTON,COOPERCRONK,CAMERONSMITHANDEITHERBILLYSLATERORGREGINGLISATFULLBACKTHEYHAVEASPINETHATISUNMATCHEDINRUGBYLEAGUETHEYHAVEPLAYEDINSOMANYHIGHPRESSUREGAMESAPRICELESSASSETGOINGINTOSATURDAYTHEKIWISAREALOTLESSEXPERIENCEDBUTWINNINGADRAMATICMATCHLIKETHEIRSEMIFINALAGAINSTENGLANDWILLDOWONDERSFORTHEIRCONFIDENCETHEYDEFEATEDAUSTRALIAINTHEFOURNATIONSFINALIN2010ANDTHELASTWORLDCUP,ANDKNOWTHEYCANRISETOTHEBIGOCCAS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