Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Chapter 6Continuous Probability Distributions8.1Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Probability Density FunctionsUnlikeadiscreterandomvariablewhichwestudiedinChapter7,acontinuous random variableisonethatcanassumeanuncountablenumberofvalues.Wecannotlistthepossiblevaluesbecausethereisaninfinitenumberofthem.Becausethereisaninfinitenumberofvalues,theprobabilityofeachindividualvalueisvirtually0.2Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Point Probabilities are ZeroBecausethereisaninfinitenumberofvalues,theprobabilityofeachindividualvalueisvirtually0.Thus,wecandeterminetheprobabilityofarange of valuesonly.E.g.withadiscreterandomvariableliketossingadie,itismeaningfultotalkaboutP(X=5),say.Inacontinuoussetting(e.g.withtimeasarandomvariable),theprobabilitytherandomvariableofinterest,saytasklength,takesexactly5minutesisinfinitesimallysmall,henceP(X=5)=0.It is meaningful to talk about P(X 5).3Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Probability Density FunctionAfunctionf(x)iscalledaprobability density function(overtherangea x bifitmeetsthefollowingrequirements:1)f(x)0forallxbetweenaandb,and2)Thetotalareaunderthecurvebetweenaandbis1.0f(x)xbaarea=14Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Uniform DistributionConsidertheuniform probability distribution(sometimescalledtherectangular probability distribution).Itisdescribedbythefunction:f(x)xbaarea=widthxheight=(ba)x=15Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.1(a)Theamountofgasolinesolddailyataservicestationisuniformlydistributedwithaminimumof2,000gallonsandamaximumof5,000gallons.Find the probability that daily sales will fall between 2,500 and 3,000 gallons.Algebraically:whatisP(2,500 X 3,000)?f(x)x5,0002,0006Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.1(a)P(2,500 X 3,000)=(3,0002,500)x=.1667“there is about a 17% chance that between 2,500 and 3,000 gallons of gas will be sold on a given day”f(x)x5,0002,0007Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.1(b)Theamountofgasolinesolddailyataservicestationisuniformlydistributedwithaminimumof2,000gallonsandamaximumof5,000gallons.What is the probability that the service station will sell at least 4,000 gallons?Algebraically:whatisP(X 4,000)?f(x)x5,0002,0008Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.1(b)P(X 4,000)=(5,0004,000)x=.3333“There is a one-in-three chance the gas station will sell more than 4,000 gallons on any given day” f(x)x5,0002,0009Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.1(c)Theamountofgasolinesolddailyataservicestationisuniformlydistributedwithaminimumof2,000gallonsandamaximumof5,000gallons.What is the probability that the station will sell exactly 2,500 gallons?Algebraically:whatisP(X = 2,500)?f(x)x5,0002,00010Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.1(c)P(X = 2,500)=(2,5002,500)x=0“The probability that the gas station will sell exactly 2,500 gallons is zero”f(x)x5,0002,00011Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.The Normal DistributionThenormal distributionisthemostimportantofallprobabilitydistributions.Theprobabilitydensityfunctionofanormal random variableisgivenby:Itlookslikethis:Bellshaped,Symmetricalaroundthemean12Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.The Normal DistributionImportant things to note:Thenormaldistributionisfullydefinedbytwoparameters:itsstandarddeviationandmeanUnliketherangeoftheuniformdistribution(axb)Normaldistributionsrange from minus infinity to plus infinityThenormaldistributionisbellshapedandsymmetricalaboutthemean13Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Standard Normal DistributionAnormaldistributionwhosemeaniszeroandstandarddeviationisoneiscalledthestandard normal distribution.Asweshallseeshortly,anynormaldistributioncanbeconvertedtoastandardnormaldistributionwithsimplealgebra.Thismakescalculationsmucheasier.01114Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Normal DistributionThenormaldistributionisdescribedbytwoparameters:itsmeananditsstandarddeviation.Increasingthemeanshiftsthecurvetotheright15Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Normal DistributionThenormaldistributionisdescribedbytwoparameters:itsmeananditsstandarddeviation.Increasingthestandarddeviation“flattens”thecurve16Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesWecanusethefollowingfunctiontoconvertanynormalrandomvariabletoastandardnormalrandomvariableSome advice: always draw a picture!017Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesWecanusethefollowingfunctiontoconvertanynormalrandomvariabletoastandardnormalrandomvariableThis shifts the mean of X to zero018Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesWecanusethefollowingfunctiontoconvertanynormalrandomvariabletoastandardnormalrandomvariableThis changes the shape of the curve019Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesExample:Thetimerequiredtobuildacomputerisnormally distributedwithameanof50minutesandastandarddeviationof10minutes:Whatistheprobabilitythatacomputerisassembledinatimebetween45and60minutes?Algebraicallyspeaking,whatisP(45 X 60)?020Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesP(45 X 60)?0meanof50minutesandastandarddeviationof10minutes21Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesOK,weveconvertedP(45 X 60)foranormaldistributionwithmean=50andstandarddeviation=10toP(.5 Z 1)i.e.thestandardnormaldistributionwithmean=0andstandarddeviation=1soWheredowegofromhere?!22Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesP(.5 Z 1)lookslikethis:TheprobabilityistheareaunderthecurveWewilladdupthetwosections:P(.5Z0)andP(0Z1)0.5123Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesWecanuseTable3inAppendixBtolook-upprobabilitiesP(0 Z z)WecanbreakupP(.5 Z 1)into:P(.5 Z 0)+P(0 Z 1)Thedistributionissymmetricaroundzero,so(multiplyingthroughbyminusoneandre-arrangingtheterms)wehave:P(.5ZZ0)=P(0 Z .5)Hence:P(.5 Z 1) = P(0 Z .5) + P(0 Z 1)24Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesHowtouseTable3Thistablegivesprobabilities P(0 Z z)Firstcolumn=integer+firstdecimalToprow=seconddecimalplaceP(0Z0.5)P(0Z1)P(.5 Z 1) = .1915 + .3414 = .532825Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Calculating Normal ProbabilitiesRecap:Thetimerequiredtobuildacomputerisnormallydistributedwithameanof50minutesandastandarddeviationof10minutesWhatistheprobabilitythatacomputerisassembledinatimebetween45and60minutes?P(45X60)=P(.5Z 1.6)?01.6P(0 Z 1.6) = .5 P(0 Z 1.6)= .5 .4452= .0548z27Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal Table (Table 3)WhatisP(Z -2.23)?02.23P(0 Z 2.23)P(Z 2.23)= .5 P(0 Z 2.23)P(Z -2.23)28Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal Table (Table 3)WhatisP(Z 1.52)?01.52P(Z 0) = .5P(Z 1.52) = .5 + P(0 Z 1.52)= .5 + .4357= .9357zP(0 Z 1.52)29Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal Table (Table 3)WhatisP(0.9 Z 1.9)?00.9P(0 Z 0.9)P(0.9 Z 1.9) = P(0 Z 1.9) P(0 Z 0.9)=.4713 .3159 = .1554z1.9P(0.9 Z 1.9)30Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.2Thereturnoninvestmentisnormallydistributedwithameanof10%andastandarddeviationof5%.Whatistheprobabilityoflosingmoney?WewanttodetermineP(X0).Thus,31Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.2Ifthestandarddeviationis10theprobabilityoflosingmoneyisP(X zA) = A33Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Finding Values of ZWhatvalueofzcorrespondstoanareaunderthecurveof2.5%?Thatis,whatisz.025?Area = .50Area = .025Area = .50.025 = .4750Ifyoudoa“reverselook-up”onTable3for.4750,youwillgetthecorrespondingzA=1.96SinceP(z1.96)=.025,wesay:z.025 = 1.9634Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Finding Values of ZOtherZvaluesareZ.05=1.645Z.01=2.3335Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Using the values of ZBecausez.025=1.96and-z.025=-1.96,itfollowsthatwecanstateP(-1.96Z1.96)=.95SimilarlyP(-1.645Z1.645)=.9036Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Exponential DistributionAnotherimportantcontinuousdistributionistheexponential distributionwhichhasthisprobabilitydensityfunction:Notethatx0.Time(forexample)isanon-negativequantity;theexponentialdistributionisoftenusedfortimerelatedphenomenasuchasthelengthoftimebetweenphonecallsorbetweenpartsarrivingatanassemblystation.Notealsothatthemeanandstandarddeviationareequaltoeachotherandtotheinverseoftheparameterofthedistribution(lambda)37Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Exponential DistributionTheexponentialdistributiondependsuponthevalueofSmallervaluesof“flatten”thecurve:(E.g.exponentialdistributionsfor=.5,1,2)38Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Exponential DistributionIfXisanexponentialrandomvariable,thenwecancalculateprobabilitiesby:39Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Example 8.6Thelifetimeofanalkalinebattery(measuredinhours)isexponentiallydistributedwith=.05Find the probability a battery will last between 10 & 15 hoursP(10X15)P(10X2.43Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Student t DistributionInmuchthesamewaythatanddefinethenormaldistribution,thedegreesoffreedom,definestheStudenttDistribution:Asthenumberofdegreesoffreedomincreases,thetdistributionapproachesthestandardnormaldistribution.Figure 8.2444Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Determining Student t ValuesThestudenttdistributionisusedextensivelyinstatisticalinference.Table4inAppendixBlistsvaluesofThatis,valuesofaStudenttrandomvariablewithdegreesoffreedomsuchthat:ThevaluesforAarepre-determined“critical”values,typicallyinthe10%,5%,2.5%,1%and1/2%range.45Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Using the t table (Table 4) for valuesForexample,ifwewantthevalueoftwith10degreesoffreedomsuchthattheareaundertheStudenttcurveis.05:Area under the curve value (tA) : COLUMNDegrees of Freedom : ROWt.05,10t.05,10=1.81246Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Student t Probabilities and ValuesExcelcancalculateStudentdistributionprobabilitiesandvalues.47Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chi-Squared DistributionThechi-squareddensityfunctionisgivenby:Asbefore,theparameteristhenumberofdegreesoffreedom.Figure 8.2748Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chi-Squared DistributionNotes:Thechi-squareddistributionisnotsymmetricalThesquare,i.e.,forcesnon-negativevalues(e.g.findingP()=A:49Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chi-Squared DistributionForprobabilitiesofthissort:Weuse1A,i.e.wearedeterminingP(0.Twoparametersdefinethisdistribution,andlikewevealreadyseentheseareagaindegrees of freedom.isthe“numerator”degreesoffreedomandisthe“denominator”degreesoffreedom.55Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.ThemeanandvarianceofanFrandomvariablearegivenby:andTheFdistributionissimilartothedistributioninthatitsstartsatzero(isnon-negative)andisnotsymmetrical.F Distribution56Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Determining Values of FForexample,whatisthevalueofFfor5%oftheareaundertherighthand“tail”ofthecurve,withanumeratordegreeoffreedomof3andadenominatordegreeoffreedomof7?Solution:usetheFlook-up(Table6)Numerator Degrees of Freedom : COLUMNDenominator Degrees of Freedom : ROWF.05,3,7There are different tablesfor different values of A.Make sure you start withthe correct table!F.05,3,7=4.3557Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.Determining Values of FForareasunderthecurveonthelefthandsideofthecurve,wecanleveragethefollowingrelationship:Paycloseattentiontotheorderoftheterms!58Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.F Probabilities and ValuesExcelcancalculateFdistributionprobabilitiesandvalues.59