离散数学概率PPT课件.ppt

DiscreteMathCS2800 Prof BartSelmanselman cs cornell eduModuleProbability Partd 1 ProbabilityDistributions2 MarkovandChebyshevBounds 1 DiscreteRandomvariable DiscreterandomvariableTakesononeofafinite oratleastcountable numberofdifferentvalues X 1ifheads 0iftailsY 1ifmale 0iffemale phonesurvey Z ofspotsonfaceofthrowndie 2 ContinuousRandomvariable Continuousrandomvariable r v TakesononeinaninfiniterangeofdifferentvaluesW GDPgrows shrinks thisyearV hoursuntillightbulbfailsForadiscreter v wehaveProb X x i e theprobabilitythatr v Xtakesonagivenvaluex Whatistheprobabilitythatacontinuousr v takesonaspecificvalue E g Prob X light bulb fails 3 14159265hrs However rangesofvaluescanhavenon zeroprobability E g Prob 3hrs X light bulb fails 4hrs 0 1Rangesofvalueshaveaprobability 0 3 ProbabilityDistribution Theprobabilitydistributionisacompleteprobabilisticdescriptionofarandomvariable Allotherstatisticalconcepts expectation variance etc arederivedfromit Onceweknowtheprobabilitydistributionofarandomvariable weknoweverythingwecanlearnaboutitfromstatistics 4 ProbabilityDistribution ProbabilityfunctionOneformtheprobabilitydistributionofadiscreterandomvariablemaybeexpressedin ExpressestheprobabilitythatXtakesthevaluexasafunctionofx aswesawbefore 5 ProbabilityDistribution TheprobabilityfunctionMaybetabular 6 ProbabilityDistribution TheprobabilityfunctionMaybegraphical 1 2 3 50 33 17 7 ProbabilityDistribution TheprobabilityfunctionMaybeformulaic 8 ProbabilityDistribution Fairdie 9 ProbabilityDistribution Theprobabilityfunction properties 10 CumulativeProbabilityDistribution CumulativeprobabilitydistributionThecdfisafunctionwhichdescribestheprobabilitythatarandomvariabledoesnotexceedavalue Doesthismakesenseforacontinuousr v Yes 11 CumulativeProbabilityDistribution CumulativeprobabilitydistributionTherelationshipbetweenthecdfandtheprobabilityfunction 12 CumulativeProbabilityDistribution Die throwing graphical tabular 13 CumulativeProbabilityDistribution ThecumulativedistributionfunctionMaybeformulaic die throwing 14 CumulativeProbabilityDistribution Thecdf properties 15 OfadiscreteprobabilitydistributionOfacontinuousprobabilitydistributionOfadistributionwhichhasbothacontinuouspartandadiscretepart ExampleCDFs 16 Functionsofarandomvariable Itispossibletocalculateexpectationsandvariancesoffunctionsofrandomvariables 17 Functionsofarandomvariable ExampleYouarepaidanumberofdollarsequaltothesquarerootofthenumberofspotsonadie Whatisafairbettogetintothisgame 18 Functionsofarandomvariable LinearfunctionsIfaandbareconstantsandXisarandomvariableItcanbeshownthat Intuitively whydoesbnotappearinvariance And whya2 19 TheMostCommonDiscreteProbabilityDistributions somediscussedbefore 1 Bernoullidistribution2 Binomial3 Geometric4 Poisson 20 Bernoullidistribution TheBernoullidistributionisthe coinflip distribution XisBernoulliifitsprobabilityfunctionis X 1isusuallyinterpretedasa success E g X 1forheadsincointossX 1formaleinsurveyX 1fordefectiveinatestofproductX 1for madethesale trackingperformance 21 Bernoullidistribution Expectation Variance 22 Binomialdistribution ThebinomialdistributionisjustnindependentBernoullisaddedup Itisthenumberof successes inntrials IfZ1 Z2 ZnareBernoulli thenXisbinomial 23 Binomialdistribution ThebinomialdistributionisjustnindependentBernoullisaddedup Testingfordefects withreplacement HavemanylightbulbsPickoneatrandom testfordefect putitbackPickoneatrandom testfordefect putitbackIftherearemanylightbulbs donothavetoreplace 24 Binomialdistribution Let sfigureoutabinomialr v sprobabilityfunction Supposewearelookingatabinomialwithn 3 WewantP X 0 Canhappenoneway 000 1 p 1 p 1 p 1 p 3WewantP X 1 Canhappenthreeways 100 010 001p 1 p 1 p 1 p p 1 p 1 p 1 p p 3p 1 p 2WewantP X 2 Canhappenthreeways 110 011 101pp 1 p 1 p pp p 1 p p 3p2 1 p WewantP X 3 Canhappenoneway 111ppp p3 25 Binomialdistribution So binomialr v sprobabilityfunction 26 Binomialdistribution Typicalshapeofbinomial Symmetric 27 Expectation Variance Aside AssumeV X V Y andXandYareindependentR Vs then But Hmm 28 Binomialdistribution Asalesmanclaimsthatheclosesadeal40 ofthetime Thismonth heclosed1outof10deals Howlikelyisitthathedid1 10orworsegivenhisclaim 29 Binomialdistribution Lessthan5 or1in20 So it sunlikelythathissuccessrateis0 4 Note So ismostlikelybutmaynotbethatlikely 30 Binomialandnormal Gaussiandistribution Thenormaldistributionisagoodapproximationtothebinomialdistribution large n smallskew B n p Prob densityfunction 31 GeometricDistribution Ageometricdistributionisusuallyinterpretedasnumberoftimeperiodsuntilafailureoccurs Imagineasequenceofcoinflips andtherandomvariableXistheflipnumberonwhichthefirsttailsoccurs Theprobabilityofahead asuccess isp 32 Geometric Let sfindtheprobabilityfunctionforthegeometricdistribution etc So xisapositiveinteger 33 Geometric Notice thereisnoupperlimitonhowlargeXcanbeLet scheckthattheseprobabilitiesaddto1 Geometricseries 34 Geometric Expectation Variance SeeRosenpage158 example17 differentiatebothsidesw r t p 35 Poissondistribution ThePoissondistributionistypicalofrandomvariableswhichrepresentcounts Numberofrequeststoaserverin1hour Numberofsickdaysinayearforanemployee 36 ThePoissondistributionisderivedfromthefollowingunderlyingarrivaltimemodel Theprobabilityofanunitarrivingisuniformthroughtime Twoitemsneverarriveatexactlythesametime Arrivalsareindependent thearrivalofoneunitdoesnotmakethenextunitmoreorlesslikelytoarrivequickly 37 Poissondistribution TheprobabilityfunctionforthePoissondistributionwithparameter is islikethearrivalrate highermeansmore fasterarrivals 38 Poissondistribution Shape Low Med High 39 MarkovandChebyshevbounds 40 Often youdon tknowtheexactprobabilitydistributionofarandomvariable Westillwouldliketosaysomethingabouttheprobabilitiesinvolvingthatrandomvariable E g whatistheprobabilityofXbeinglarger orsmaller thansomegivenvalue Weoftencanbyboundingtheprobabilityofeventsbasedonpartialinformationabouttheunderlyingprobabilitydistribution Partialinfo e g theexpectation Why MarkovandChebyshevbounds 41 Theorem MarkovInequality LetXbeanonnegativerandomvariablewithE X Then foranyt 0 Hmm Whatif Sure Note relatescumulativedistributiontoexpectedvalue gives But Can thavetoomuchprob totherightofE X 42 Proof WheredidweuseX 0 3rdline I e 43 Alt proof MarkovInequality Define E Y E X 44 Example Considerasystemwithmeantimetofailure 100hours UsetheMarkovinequalitytoboundthereliabilityofthesystem R