Summary Notes 3

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Irvine Valley College**We aren't endorsed by this school

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CS 120

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Mathematics

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Dec 16, 2024

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SummaryNotes3-Calculus+ProbabilityDS120SpenJanimaChatterje

PROBABILITY1.whenordermatters:i.e.whenEA,B,Cis&differentfromEB,A,c}eg.HATisadifferentwordwhileAHisadifferentwordwiththesameletters.-UsePermutation:upM:r)!⊥-Totaloptionsavailable=r-Totalemptyplacestofullmany6letterwords(withoutletterEg.Howmakewithrepeated)canyoubeinglettersofthealphabet?all26-n=26:r=6(26-)!orderdoesn'tmatter,weuseCombination2.whenwhenEA,B,CYisthesameasEB,A,C.i.2..---Combination:r=T

Eg.Agrouphasspeople.Inhowmanydifferentwayscanyoumakeacommitteeof3peopleoutofthisgroup.->n=5M=3Orderdoesn'tmatter:Combination.3Cz·Probabilityofaneventhappening:#ofwaystheeventcanhappen#ofoutcomes·Independentevents:OccuranceofIeventdoesn'teffecttheocuranceoftheotherevent.·Probabilityof2independenteventsP(AandB)=P(A).P(B)·Disjointevents:Eventsthatcannothappentogether·P(AorB)=P(A)+P(B)

·Binomialdistribution:eventswith2outcomes-Worksforp:probabilityofoneoutcome7&a:probabilityofotheroutcome:C-p)N:Totalnumberoftrialsr:Totalnumberofsuccesses(happeningoftheareinterestedin)eventwe-ProbabilityofobervingrsuccessesinNindependenttrials:N-Mp(X=r,N,p)=NCq(-p)CONDITIONALPROBABILITY-·probabilityofeventA,guiereventB:P(ArB)P(AIB)-P(B)-AandBareindependent:IP(A(B)=P=P(A)PB)OccuranceofBdoesn'teffectoccuranceofA·Bay'sRule:P(B(A)=p(A1B)·P(B)-p(A)

#ECTATIONP(Xx)X=x_probabilitywewillmultiplytheindividuali.2.ofeachoutcometotheoutcomeandseemitforalloutcomes·E [X+Y]=E(X]+E[y]·ETaX]=aE[X]VARIANCE-·van(x)=E((X-E(xi)]i.e..youcalculateexpectedvalue.-F(X)expectedvaluefromeachoutcome2·subtractsquarethem->(n-E(X])=and-eachtermtotheirindividual*multiplyTp(X=u)(x-E[x]]probabilitytermstogether·Addallthe&(a-E[X])?.P(X=x)X=x·Var(X+Y)=Var(X)+Var(Y)·Var(aX)=Var (X)

GAUSSIANANDCENTRALLIMITTHEOREM#--Meanofsample--/populationM-MeanoforiginaldistributionIIMy-Meanofsamplingdistributionofsamplemean-Istandarddeviationofsample⊥IstandarddeviationofsamplingdistributionofsamplemeansMX=UissamplesizewheretU=Onz:M=X-M-Ex/95%ofthedatalis1.91standarddeviationsawaymeanfromthez-Tellsusnowmanystandarddeviationsawayfromtheactualmean,ourobservedmeanis

⊥-StandardError&"-varianceofsampleu=w<varianceofsamplingdistributionofsamplemeans&ConfidenceInterval:Rangeinwhichourtruemeanislikelytoliewithacertainprobability:&M=Ex-z.8,X+28x)*-Observedmeancorrespondingtoconfidence/probabilityz-ValuefromConfidenceTable.(Ifnothingisgiven,assume95%ConfidenceU-standarderror

#InordertocalculateConfidenceInterval,_a*(observedsampleMean)1.IdentifyIdentifyorQ(varianceorstandarddeviation2.-oestimatedbytheoforiginaldistribution8samplevarianceorsamplestandarddeviation)3.Identifyn,(samplesize)1.Calculatestandarderror-by=nXD--Confidencepercentage5.Tentify&assume95%](Ifnothingisgiven,z-valueofthe6.LookupE-tableforgivenconfidencepercentageF.CalculateConfidenceInternal:m:[5-b·Y+z.=]z-

BESSEL'SCORRECTIONIfsampledataisgivenandyouhavetocalculatethevariancefromthedata,useBessel'sCorrectiontocalculateMakesureyouEx2g[-*(xi-x)n-1i=1⑤degreesoffreedomIfsamplesize30:Noz-tableIfsamplesize30:Uset-table

whensamplesize30->usetdistributionwhensamplesize>30-usezdistributionwhenVariance/S.d.Notgiventous,-toestimateBessel'scorrection.usedeirationfromthepopulationstandardwhenn30(preferablesample-Especiallybesselsifsampleisthetoalwaysapply-usedto*interval:estimate.-acalculatingconfidencepopulationForsid.SIsYgiven?IIYes-No*P-actualpopulationJeS.&.CalculateX-IIS-estimatedpopulation-s.d.Isusquien?ITNoIYesI↓svessel'scorrection-↑CalculateusingSorUx=&-v⊥standarderror.I.2.Op=MnCalculate#Isn 30-NoIYes*useM=Y+z.ExUseM=Y+t.U

MULTIVARIABLECALCULUS·f(x)=an-f'(x)=nx"-·h(x)=fa+g(x)+h'(x)=y'(x)+g(x)·h(x)=f(x).g(u)->h'(x)=y'(x).g(x)+g(x).f(x)·hx=f (g(x)->hea)=y (g().g'()-Partialderivatives,Multivariablefuncions:forSymbol/Notation-Oinsteadofd·I-Partialderivativewithrespecttox.Treatothervariableslikeanyothernumber/constant·Tny-Orderofvariablesinthesubscripttellyoutheorderinwhichthepartialderivativewascalculated

Howtofindsaddlepoints,relativeMaximumorMinimum?-Calculategradientoff(x,y,z...)·+8 ]->EquateIfto0:If=0-Calculatethevalueof(R,Y,z...)forwhichIfcriticalpointsisO.Theseareyour2->CalculateD=YouDyy-buyeachcriticalpoint-EvaluateDforJSaddlepoint·130-·IfDoandJunLo-RelativeMax·IfDoandfun>0-RelativeMin

DIRECTIONALDERIVATIVES·DirectionalDerivativeoff(K,Y,2)atapoint(No,Yo,zo)alongaunitvector:**f=If(4,%2)·isf*unitvectoriinthedirectionof-·Directionofstepestdescent:-f·Directionofstepestascent:Of·onestepofgradientdescent:xj=no-&Tf(x)d+stepsizeIngeneral,en=an-def(um)y=%.-d-f(yo)IngeneralYn+1=Yn-d+f(yn)

IFredislivinguphisfourgoltrophiesonaNshelf.Howmanydifferentpossiblearrangementscanhemake?passwordconsistsof3digits,oto9),followed&Afromthealphabet(25letters).Ifbylettersbutrepetition10-arepertonofdigitsisallowed,differentisnotallowed,howmanyofalphabetspasswordscanbemade?

&Astudentcalculatesa90%confidenceintervalI--forMwhenoisknown.Theconfidenceintervalcentsto64.3cents.Whatisis-24.3thesamplemeanX?IntheOhioLottery,thereisagamecalled-Q."Lucky4".Theplayerpays$10topickaIdigitnumber(repetitionsallowed).IfintheLottery,4numberscomeupinthe&expected,Youwin$1000.orderasyoujackedsamevalue?Whatisyour

OFindallthefirstorderpartialderivatives-yofandvalculatethegradientat(1,1)f(x,y)-3x3y+4ySATscoresarenormallydistributedwithmean&.istsandstandarddeviation325.aIfonesatscoreischosenatrandom,findtheprobabilitythatitisbetweenand14801440b.If16SATscoresarechosenatrandom,findtheprobabilitythattheyhaveameanbetween1440and1480.

aCalculatethedirectionalderivativeof-f(x,y)=3x2+47inthedirectionofvectorT:(i]atpoint(2,)

orKneeQ.200patientswhohadeitherchipsurgerysatisfied-weresurgerywereaskedwhethertheyordissatisfiedregardingtheirresults.surgerysatisfiedDissatisfiedTotalKnee702595Hip9015105Total16040200these200isselectedat-3a)Ifonepersonfromprobabilitythepatientrandom,determinetheissatisfied,giventheyhadunlesurgeryb)wasdissatisfiedgiventheyhadhipsurgery