概率统计英文讲义Walpol.ppt

Probability StatisticsforEngineers Scientists byWalpole Myers Myers Ye Chapter2Notes ClassnotesforISE201SanJoseStateUniversityIndustrial SystemsEngineeringDept SteveKennedy ProbabilityIntro ThesamplespaceSofanexperimentisthesetofallpossibleoutcomes Wemustunderstandthesamplespaceinordertodeterminetheprobabilityofeachoutcomeoccurring Thesamplespaceisaset thedomainoftheprobabilityfunction Eachprobabilityvalue p isarealnumber0 p 1 Important Thesumoftheprobabilitiesforallelementsinthesamplespacealwaysequals1 Whyisthisimportant Thisfactallowsustocheckouranswers Properlyenumeratingthesamplespaceiskeytocorrectlycalculatingprobabilities Atreediagramissometimesuseful Events Aneventisasubsetofasamplespace E S NotethatbothSand areeventsaswell Samplespacescanbecontinuousordiscrete Whatisacontinuousvs discretesamplespace Example Lifeinyearsofacomponent S S t t 0 allvaluesoftsuchthatt 0 A componentfailsbeforetheendofthefifthyear A t t 5 Example Flipacointhreetimes S S HHH HHT HTH HTT THH THT TTH TTT EventA 1stflipisheads A HHH HHT HTH HTT Event SetOperations ThecomplementofaneventA ThesetofallelementsofSnotinA DenotedA A 1stflipisheads A firstflipisnotheads TheintersectionoftwoeventsAandB ThesetofallelementsinbothAandB DenotedA B B 2ndor3rdflip butnotboth areheads B HHT HTH THT TTH A B A B HHT HTH Twoeventsaremutuallyexclusiveif A B Theunionoftwoevents AandB ThesetofelementsineitherAorB A B A B HHH HHT HTH HTT THT TTH VennDiagrams VennDiagramsshowvariouseventsgraphically andaresometimeshelpfulinunderstandingsettheoryproblems Standardsettheoryresultshold A A A A A A S A A B A B A B A B A B A B IntuitiveSamplePointCounting Ifoneoperationcanbeperformedinn1ways andforeachway asecondcanbeperformedinn2ways thenthetwocanbeperformedatotalofn1n2ways Forthreeoperations n1n2n3 Howmanypasswordsoflength5needtobecheckedbyapasswordhackingprogramifonlylowercaselettersareused 265 11 881 376 Thisiscalledsamplingwithreplacement Atreediagramcanbeusedtoenumeratealloftheoptions PermutationOrderings Apermutationisanorderingofasetorsubsetofobjects Thenumberofdistinctorderingsofnitems nitemscangointhefirstposition Oncethefirstitemisfixed n 1itemscangointhe2ndposition Thenn 2itemsinthethirdposition etc Numberoforderingsisn n 1 n 2 1 orn Rememberthat1 0 1 Thisiscalledsamplingwithoutreplacement Onceweuseavalue itcan tbeusedagain PermutationsofPartialOrderings Supposethatwewillgive3differentawardstothreestudentsoutofaclassof60students Howmanywayscantheawardsbegiven Whatiftheproblemwasslightlydifferentandonestudentcouldwinallthreeawards Thiswouldbewithreplacement Thenumberis603 216 000Withoutreplacement 60studentscouldgetthefirstaward then59studentsareeligibleforthe2ndand58forthethird or60 59 58 205 320 Ingeneral thenumberofpermutationsofnthingstakenratatimeiswrittennPr n n r 60 57 60 59 58 OtherPermutations Circularpermutations ndistinctobjectsarrangedinacircle Thepositionofthefirstobjectcouldbeanywhere Ifallobjectsmovedonepositionclockwise it sstillthesamepermutation Fixthefirstobjectanywhereonthecircle thenn 1objectscangototheleft n 2 next etc Thenumberofcircularpermutationsofnobjectsis n 1 Anotherwaytolookatit thesetofallpermutationsisn Foreachstartingvalue therearenorderingsthatareidentical movingthesameorderingaroundthecircle Sothetotalnumberofdifferentorderingsisn n n 1 Forexample fortheordering35142forn 5 Thereare5identicalorderings 35142 51423 14235 42351 23514 PermutationswithIdenticalObjects Ifsomeobjectsareidentical withn1oftype1 n2oftype2 nkoftypek andn n1 n2 nk thenumberofdistinctpermutationsisn n1 n2 nk Example 3itemsoftype1 1oftype2gives4permutations 1112 1121 1211 and2111 Foreachofthefour therewouldbe3 orderingsifthe1 sweredistinct saya b andc Forexample 1121wouldbeab2c ac2b ba2c bc2a ca2b andcb2a Or ifallitemsaredistinct therewouldbe4 3 Or4 orderings Threeitemsbeingidenticalreducesthenumberofpermutationsbyafactorof3 sowedivideby3 ArrangingnObjectsIntorCells Partitioningndistinctobjectsintorcellsorsubsets eachofagivenfixedsize wheretheorderingofobjectswithinacelldoesn tmatter Exampledivide5itemsintotwocells oneofsize3andoneofsize2 1 2 3 4 5 1 2 4 3 5 1 2 5 3 4 1 3 4 2 5 1 3 5 2 4 1 4 5 2 3 2 3 4 1 5 2 3 5 1 4 2 4 5 1 3 3 4 5 1 2 Therearen totalpossibleorderings butn1 Inthe1stcell andn2 inthe2ndcell etc areidentical Ingeneral thenumberofdistinctcombinationsofndistinctobjectsintorcells withn1itemsinthe1stcell n2inthe2nd andnrintherthcellisn n1 n2 nr CombinationsofnItemsTakenrataTime Toreview therearehowmanypermutationsofnitemstakenratatimeifeachorderingisdistinct n n 1 n 2 n r 1 ornPr n n r Foranygivensetofritems therearer possibleorderings Sowhatiftheorderoftheritemsdoesn tmatter DividenPrbyr togetthenumberofdistinctoutcomes Thenumberofcombinationsofnitemstakenratatime whereorderdoesn tmatter isnCr n r n r Example Howmanydistinctpokerhandsof5cardseachcanbedealtusingadeckof52cards 52C5 52 5 47 52 51 50 49 48 5 4 3 2 1 2 598 960 ProbabilityofanEvent Fornow weonlyconsiderdiscretesamplespaces Eachpointinasamplespaceisassignedaweightorprobabilityvalue Thehighertheprobability themorelikelythatoutcomeistooccur TheprobabilityofaneventAisthesumoftheprobabilitiesoftheindividualpointsinA Then 0 P A 1P P S Iftwoeventsaremutuallyexclusive whichmeans thattheyhavenopointsincommon orA B thenP A B P A P B RelativeFrequencyforProbability IfanexperimenthasNdifferentequallylikelyoutcomes andnoutcomescorrespondtoeventA thenP A P A n N AdditiveProbabilityRules WealreadyknowthatifAandBaremutuallyexclusive P A B P A P B Also ifmorethan2eventsaremutuallyexclusive theprobabilityoftheunionofallofthoseeventsisthesumofalloftheindividualprobabilities WhatisP A B ifAandBarenotmutuallyexclusive CanuseaVenndiagramtoshowthiscase ThesamplepointsinP A B aredoublecounted SoP A B P A B P A P B P A B GiventhatweknowP A whatisP A P A 1 P A ConditionalProbability Conditionalprobability writtenP B A istheprobabilityof B givenA theprobabilitythatBoccurs giventhatweknowthatAhasoccurred LookataVenndiagramofAandB Supposeallsamplepointsareequallylikely P B ofoutcomesinB total ofoutcomesinS P B A P B A ofoutcomesinA B total ofoutcomesinA or P B A P A B P A aslongasP A 0 ConditionalProbabilityExample Example Forthefollowingpopulationof900people EmployedUnemployedTotalMale360140500Female240160400Total600300900Ifapersonisselectedatrandomfromthisgroup P E P M P E M P E M P M E Answers P E 600 900 2 3 P M 500 900 5 9 P E M 360 900 2 5 P E M 360 500 18 25 P M E 360 600 3 5 Independence Conditionalprobabilityhelpsusupdatetheprobabilityofaneventgivenadditionalinformation SupposeP B A P B Whatdoesthistellus WhetherAoccursornot theprobabilityofBoccurringdoesn tchange IfP B A P B thenAandBareindependent CanshowthatifP B A P B istrue thenP A B P A isalwaysalsotrue Fromtheabove andthedefinitionofconditionalprobability ifAandBareindependent P A B P A P B MultiplicativeRules Rearrangingtheconditionalprobabilityformula ifbothAandBcanoccur thenP A B P B A P A Or theprobabilityofbothAandBoccurringequalstheprobabilityofBgivenAtimestheprobabilityofA NotethatitisalsotruethatP A B P A B P B If andonlyif eventsAandBareindependent thenfromtheaboveformula wehaveP A B P A P B Formorethantwoindependentevents multiplyalloftheprobabilitiestogether TheoremofTotalProbability SupposethesamplespaceScanbepartitionedintoeventsA1 A2 andA3 Whatdoesthismean A1 A2 andA3aredisjointandbetweenthemcoverallofS ThentheprobabilityofaneventBoccurringcanbecalculatedusingconditionalprobabilitiesgiventhateitherA1orA2orA3occurred P B P B A1 P A1 P B A2 P A2 P B A3 P A3 Inwords thismeans TheprobabilitythatBandA1occur theprobabilitythatBandA2occur theprobabilitythatBandA3occur Thisrule calledthetheoremoftotalprobability ortheruleofelimination holdsforanypartitioningofS BayesRule Herewecancalculatereverseconditionalprobabilities Usinganexamplefromanearl