CHAPTER 3: Random Variables and Probability Distributions

1 CHAPTER3: RandomVariablesandProbability DistributionsConceptof a Theoutcomeof a randomexperiment neednotbe a number. However,we areusuallyinterestednotin theoutcomeitself,butratherin somemeasurement of :Considertheexperiment in which batteriescomingo anassemblylinewereexamineduntil agood one(S)was , FS,FFS,: : :g:We may be interestedin thenumber of batteriesexaminedbeforetheexperiment randomvariableis a functionthatassociatea realnumber witheach element in :Tossingtwo coinsS=fHH,TT,HT,THgLetX= # of :A groupof 4 components is knownto contain2 thetimeuntil the2 defectives of thetestonwhich theseconddefective is types of randomvariables Adiscreterandomvariableis a randomvariablewhosepossiblevalueseitherc onstitutea nitesetor elsecanbe listedin anin nitesequence.

2 A randomvariableiscontinuousif itssetof possiblevaluesconsistsof anentireinterval onthenumber randomvariables,such as weight of anitem,lengthof lifeof a motoretc.,canassumeany valuein massfunctionof a discreterandomvariableXis de nedbyf(x) =P(X=x)Example:tossingtwo coinsX= # of (0)=P(X= 0) =P(TT)= 1=4f(1)=P(X= 1) =P(HT,TH)= 1=2f(2)=P(X= 2) =P(HH) = 1=4 Example:Aninformationsourceproducessymbo lsat randomfroma ve-letteralphabet:S=fa; b; c; d; eg:Theprobabilitiesof thesymbolsarep(a) =12; p(b) =14; p(c) =18; p(d) =p(e) =116:A datacompressionsystemencodesthelettersin to binarystringsas follows:a1b01c001d0001e0000 LettherandomvariableYbe equalto thelengthof thebinarystringoutputby (1)=P(Y= 1) =f(2)=p(Y= 2) =f(3)=p(Y= 3) =f(4)=p(Y= 4) =f(x) =P(X=x) satis (x) (x) = 1 Example:A box contains5 ballsnumbered1;2;3;4.

3 And5. Threeballsaredrawnat randomandwithoutreplacement themedianof thenumbersonthe3 chosenballs,thenwhatis theprobability functionforX, wherenonzero?Solution2 Example:Determinecso thatthefunctionf(x) canserve as theprobability massfunctionof a randomvariableX:f(x) =cxforx= 1;2;3;4;5 Solution:Thecumulative distributionfunction:F(x) of a discreterandomvariableXwithprobability massfunctionf(x) is de nedforeverynumberxbyF(x) =P(X x) =Xt xf(t)Example:Assumethatf(2)=p(X= 2) = 1=6f(3)=p(X= 3) = 1=3f(4)=p(X= 4) = 1=2 ThenF(2)=F(3)=F(4)=F(x) =Example:A of linesin useat a speci massfunctionofXis given belowx0123456p(x)0:100:150:200:250:200:0 60:04a. Findthecumulative distributionfunctionb.

4 Findtheprobability thatfat most3 linesarein Findtheprobability thatfat least4 linesarein :IfXhasthecumulative distributionfunction:F(x) =8>>>> <>>>>:0ifx <11=3if 1 x <41=2if 4 x <65=6if 6 x <101ifx 10; ndtheprobability : density curve is a curve that is always onor above thehorizontalaxis,and hasareaexactly1 density curve describes theoverallpatternof a andabove anyrangeof valuesis theproportionof allobservationsthatfallin :Thefunctionf(x) is aprobability density functionforthecontinuousrandomvariableX, de nedoverthesetof realnumbersR, (x) 0, 1f(x)dx= (a < X < b) =Rbaf(x) onp. 73:Theproportionof peoplewhorespondto a certainmail-ordersolicitationis a continuousrandomvariableXthathasdensity functionf(x) = 2(x+2)5if 0< x <10otherwise;(a)Show thatP(0< X <1) = 1(b)Findtheprobability thatmorethan1=4 butfewer than1=2 of thepeoplecontactedwillrespondto thistype of density(x)$xdensity(x)$ : Density histogramforthedatain Ex9 onp.

5 215 ExampleThemileage(inthousandsof miles)thatcarownersgetwitha certainkindof tireis a randomvariablehavingthedensity functionf(x) = 120e x=20ifx >00ifx 0;Findtheprobabilitiesthatoneof thesetireswilllast(a)at most10;000miles;(b)anywherefrom16;000to 24;000miles;(c)at least30; :Example:Thepdf of thesamplesof speech waveformsis foundto decay exponentiallyat a rate , so thefollowingpdf is proposed:f(x) =ce jxj 1< x <1 FindtheconstantC, andthen ndtheprobabilityP[jXj< ].SolutionExampleSupposethereactiontempe ratureXin a certainprocesshasa uniformdistributionf(x) = 110;if 5 x 5;0otherwise;ComputeP(X <0),P( 2< X <3),andP( 7< X <1)Solution:Example:Thecumulative distributionfunctionof checkouttimedurationXisF(x) =8<:0ifx <0;x24if 0 x <2;1ifx 26(a)Usethisto computeP(X 1) andP(0:5 X 1)(b)Findthedensity functionofXSolutionReview:1.

6 A saleengineerfora manufacturerof high-speedgrindingequipment hasjustreturnedfromvisiting thatthefollowingtabledescribes thedistributionof thenumber of salesshewillmake:x(#ofsales)012345P(X=x) 0:050:30:30:20:10:05a. Computeandplotthecumulative functionF(x) =P(X x).b. Findtheprobability thatthesalesengineerwillmake morethan3 Findtheprobability thatshemakes at least2 a chemicalprocessis theconcentrationdriftsoutsidethelimits,t heprocessis shut thenumber of timesin a given weekthattheprocessis theoftimesin a given weekthattheprocessis thecumulative functionF(X),F(x) =8>>>>>> <>>>>>>:0ifx <00:17if 0 x <10:53if 1 x <20:84if 2 x <30:97if 3 x <41ifx 4a. Whatis theprobability thattheprocessis re-calibratedfewer thantwo timesduringa Whatis theprobability thattheprocessis re-calibratedmorethanthreetimesduringa week?

7 C. Whatis theprobability thattheprocessis re-calibratedexactlyonceduringa week?d. Whatis theprobability thattheprocessis notre-calibratedat allduringa week?e. Whatis themostprobablenumber of re-calibrationsto occurduringa week? cationscallforthethicknessof aluminumsheetsthatareto be madeinto cansto be between8and11 thousandsof aninch. LetXbe thethicknessof densityofXis given byf(x) = x546< x <120otherwisea. Whatproportionof sheetswillmeetthespeci cation?b. Findthecumulative distributionfunctionof Findthe10thpercentileof A particularsheetis 10thousandthsof aninch thick. Whatproportionof sheetsareticker thanthis? collegeprofessornever nisheshislecturebeforethebellringsto endtheperiod andalways nisheshislectureswithin2 andtheendofthelectureandsupposethatthecu mulative distributionisF(x) =8<:0x 0x380 x 21x 2a.

8 Whatis theprobability thatthelectureendswithin1 minof thebellringing?b. Whatis theprobability thatthelecturecontinuesbeyondthebellforb etween60and90sec?c. Whatis theprobability thatthelecturecontinuesforat least90 secbeyondthebell?10 QuestionsfromOldExam1. Considera randomvariableXwiththefollowingprobabili ty massfunctionx 3012f(x):2:3:4cFindP(X >0:2).(hint: youneedto ndc rst).(a):4(b):8(c):7(d) o ersitspolicyholdera number of di erent premiumpayment randomlyselectedpolicyholder,LetX= thenumberof monthsbetweensuccessive distributionfunctionofXis as follows:F(x) =8>>>>>> <>>>>>>:0x <10:301 x <30:403 x <40:454 x <60:606 x <121x 12:ComputeP(3 X 6)(a)0:3(b)0:2(c)noneof theabove(d)R63F(x)dx3.

9 Theprobability density functionof thetimeto failureof anelectroniccomponent in a copier(inhours)isf(x) =8<:13000e x=3000x >00otherwise:Determinetheprobability thata component failsin theinterval from1000to 2000hours.(a)0:2031(b)13000(e 1=3 e 2=3)(c)0:2835(d)noneof theabove4. Supposethecumulative distributionfunctionof thelength(inmillimeters)of computercablesisF(x) =8>>>> <>>>>:0ifx 12000:1x 120if 1200< x 12101ifx >1210:Which of thefollowingstatement is true:(a)P(1000< X <1208)= 0:8(b)F(x) is nota cumulative distributionfunctionsinceF(1000)= 20(c)P(1000< X <1208)6=P(X <1208)11xF(x) :F(x)(d)P(X= 1208)= 0:85. LetXbe a :2f(x)0:40:30:3(a)FindP(X= 3jX 1)(b)FindtheCumulative distributionfunctionF(x) forall 1< x <1(c)FindP(X <4:2).

10 6. Thecumulative distributionfunctionofXis as follows:F(x) =8>>>> <>>>>:0x < 31=7 3 x <36=73 x <71x 7(a)FindP(X 1)(b) For thecumulative distributionfunctionshownin Figure2, whattype of randomvariableisX? (contin-uousor discrete).ExplainJointProbabilityDistrib utionsOnlythediscretecaseFor a givenexperiment, we areofteninterestednotonlyin Probability distributionfunctionsof individualrandomvariablesbutalsoin therelationshipbetweentwo or : In anexperiment into possiblecausesof cancer,we might be interestedin therelationshipbetweentheaveragenumber of cigarettessmoked dailyandtheageat which anindividualcontractscancer. Anengineermight be interestedin therelationshipbetweentheshearstrengthan dthediameterof aspot weldin a two discreterandomvariables,f(x; y) =P(X=x; Y=y)Example:Supposethat2 batteriesarerandomlychosenfroma groupof 3 new,4 usedbutstillworking,and5 defective we letXandYdenote,respectively, thenumber of newandusedbutstillworkingbatteriesthatar echosen,thenthejoint massfunctionofXandY,f(i; j) =P(X=i; Y=j) isgiven byf(0;0) = 52 122 =1066f(0;1) = 41 51 122 =2066f(0;2) = 42 122 =666f(1;0) = 31 51 122 =1566f(1;1) = 31 41 122 =1266f(2;0) = 32 122 =366yRowf(x; y)012totalsj010=6620=666=6636=66xj115=66 12=6627=66j23=663=66 Columntotals28=6632=666=661De :Thefunctionf(x.)