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andis equalto 0 otherwise.The Problems 247 constantB(a, ,) is givenby tl B(a, b.) x"-t1l - ry-t 'fhe ot Jo meanandvariance ofsuc nu= random Etwt 't^t=' 4 ;;i PRoBtEMs variableare Var{xr= .-E--.-.-.__\-,u- rD ) ' t ar b + t ) l. Let X be a randomyariabie witl l probability densityfuncdon Irxt= 1a) w,hatis thevalue ofc., lc{l --r2) -l <x.l otherwise ,. ; runcrion orx? otL me ".,l'J1;.,$,TTi,jjl:^9::,11,,'on x tt't'"0.""'ii *' ii " "i J,lJii;tln:jiil:f,il,tl"." rorarandom amounr fzt = Icrc-'/2 x >-o that the sysrem ,. J#i,'.'":Tflix:tJ:'jr ,"",.", .. monrhs? ",i".0,,5 I r x t = l c r 2 x - L c 3 1g ' x - ! ;::llf " aprobabiiitv a"n,rrv rl,"n",ronr ,r.".l',i,". .o. "{"n' f al = lctz' . '2) 0 *' ' " othe.ruire2 if /{") t"t" 4' Theprobabiiiry d.nri,u0,.^,,^- ^ll ofx, therifetime trneasu.eo rnrou.st,;;;*;tT ofa cerrainrype ofejecrronic device tat={# .-to t a t F i n dp l x - ) ^ t lo r 5 lo lli ,yffli f:;,TlH;t;,i,:l:r'* orx? nncrion h'."., s. An*:::1^1: "r',i'i"",'ll,l"'JL'il:';ff;J;."r reasr r wirrruncrion fora, d;;"..#g;;'11;'";i*'"il"[i"*lr#],lil voi,, me orvh.;n nnk,] ' fr'r= /s't -xf o'x'l orher*ise or rh"runkoe ',.u.rcu liil#:r:i: "apacirr ,, so that the u a given weekis .0l? probabiriryof the suppry,rbeing ex- 248 chapter 5 continuousRandomVariables 6. ComputeE[X] if X hasa densityfunctiongivenby l!'"'/2 ^'o (a, /(r):14 [o : o*rerwrse lc(l-.^2) - l<,r< I. olherwise l0 f5 .r>) l; (c) /(r) : .{' | - r: 5 [o 7. The densityfunctionof X is givenby lD) JI(r- :r:l f @:16*b',ootherwise If EtXl = l, 6nd a andD. probability 8. The tifetimi in hours of an electonic tube is a rundom variable having a density ' functiongivenbY r-0 f Qc)--xe-' Computethe expectedlifetime of such a tube. is a con9. ConsiderExample4b of Chapter4, but now supposethat the seasoraldemand the optimal that Show function / tinuous random variable having probability density satisf,es amountto stock is the value s* that F6\ =:- t) f ( the cumulatrve where, is netpront per unit sale,I is thenet lossper unit unsold,andF is demand' seasonal of the function distribution intervals starting 10. Trains headedfor destination A arrive at the hain station at 15-minute intervalsstarting l5-minute at at 7 A.M., whercasbains headedfor destinationB arrive at 7:05A.M. (a) If a certain passengerarrives at the station at a time unifonnly distributed between of time 7 and 8 e.v. and then gets on the first train that ardves, what proportion doeshe or shego to destinationA? (b) Whatif the pasiengerarrivesat a time uniformly distributedbetween7: l0 and8:l0 A.M.? statementand 11. A point is chosenat randomon a line segmentof length t lnterpret.this less than i is segment ;nger the to sfiorter oiihe ratro that ttre probability find the . 12.AbustravelsbetweenthetwocitiesAandB,whicharel00milesapart.Ifthebushasa over (0' breakdown,the distanceftom the breakdownto city A has a uniform distribution route between of the center in the B' and A, in city 100).Thereis a busseNicestationin that it would be moreeffrcientto havethe threestationslocated A and B. It is suggested ftom A' Do you agree?Why? respectively, milis, 75 25, 50, and at sometlme 13. You arrive at a bus stop at 10 o'clock' knowing that the bus will arive uniformly distributedbetweenl0 and 10:30' (a) What is the probabilitythat you will haveto wait longerthan l0 minutes? (b) If at l0:15 the bushasnot yet airived,what is the prcbabilitythat you will haveto wait at leastan additionall0 minutes? |4,LetXbgauniform(0,].)randomvariable.comPuteE[x,]byusingPloposition2.l andrhenchecl theresultb1 usingthedefinitionof expectation' Problems 249 a probability hecumulative rvals starting rvals starting rutedbetween ortionof time 7 : 1 0a n d8 : 1 0 statementand s than i. fthe bushasa [tion over(0. oute between ltions located at sometime utes? u will haveto Dposition2.1 15. If X is a normalmndomvariablewith parameters = 1,r 10 ando2 : 36, compute (a) P{X > 5}: (b) P{4<X<16}: (c) P{x < 8}; (d) P(x < 201; (e) P{X > 16). 16. The amual rainfall (in inches) in a certain region is normally distributed with lt, : 40 and q : 4. What is rhe probability that starting with thi, y""r, it *ili tut" over 10 years before a year occurshaving a rainfall of ovei 50 ln"nesf Wtat urru_puon" u." you making? 17. A man aiming at a target receives 10 points if his shot is within I inch of the target, 5 pointsif it is between1 and 3 inchesof the target,and points 3 if it is between3 and 5 inchesof the target.Find the expectednumberof points scored if the distancefrom the shot to the target is uniformly distributed betweenb and 10. 18. Supposethat X is a normalrandomvariablewith rnean5. p{X If > g} = .2, approxi_ matelywhat is Var(X)? 19. L€t X be a normal randomva.iablewith mean 12 and variance 4. Find the value of c suchthat PIX > c] = .10. 20. If.65 percent of the population of a large conmunity is in favor of a proposed rise in schooltaxes,approximatethe Fobability that a randomsanple of I00 peoplewill con_ tarn (a) at least50 who are in favorof the proposition; (b) between60 and 70 inclusive who are in favor: tc) fewer rhan75 in favor 21. supposethattheheight,in_inches, of a 25-year-oldmanis a normalrundomvariabrewith paxametersg. : '1I ando2 = 6.25. What percentageof 25_year_otJ men are over 6 feet 2 inchestall? what percentage of men in the 6-fooierclub a.e ou". e ioot s inctr"sr -22. The width of a slot of a dumluminforgingis (in inches) normallydistriOuteawith pr.= .9000ando : .0030.The specificationlimits weregivenas .SOOb j OOSO. (a) What percentageof forgings will be defective? (b) What is the maximumallowablevalueof o that will permit no more thar 1 in 100 defectivesWhenthe widthsare normallydistributedwith p = .9000 ando? 23. One thousandindependentrolls of.a fair die will be made, Compur" an upproxlmatron to the probabilitythat number6 will appea{berween150 and 20b dmesrnclusively.If number6 appearsexactly200 times,find the probabilitythat number5 will appearless than 150times. 24. The lifetimesof interactivecomputerchips producedby a certainsemlconductor man_ are n-ormally distributed with parametersp = 1.4 x. 106hours and o : :rac,r:le: r x I uwhat rstle approximateprobabilitythat a batchof 100chipswill -n-ouls. contain at least20 whoselifetimesare lessthanl.g x 106i 25. Each item produced by a certain manufacturer is, independently, of acceptablequality with probabiiiry.g5. Apprcximatethe probab ity that at most iri qith" ne*t tso itern" produced arc unacceptable. 26. Two_types of coins are produced at a factory: a fair coin and a biasedone that comesup heads55 percentof the time. We haveone of thesecoinsbut do noifiow wtretherit is a fail coin or a biasedone. In orderto ascertainwhich type of coin we have,we shall peform the following statisticartest:we sharltossthe coin 1000times.rfthe coi,' lands on heads525 or more times,thenwe shall concludethat it is a biasedcoin, whereas,if it landsheadslessthan 525 times,then we shall concludethat it is the tair coin. Ifthe 250 Chapter5 continuousRandomVariables coin is actuallyfair, what is the probabilitythat we shallreacha falseconclusion?What would it be if the coin were biased? tossesof a coin,the coin landedheads5800times ls it reasonable 27. In 10,000independent to assumethat the coin is IIot fair? Explain. 28. Twelve percent of the population is lefthanded. Approximate the probability that there are at least20 lefthandorsin a schoolof200 students.Stateyour assumptions. 29. A model for the movementof a stock supposesthat if the presentpdce of the stock is s. then after one time pedod it wilt be eitherus with Fobability p, or ds with Fobability I p. Assuming that successivemovementsarc independent,apForimate the probability that the stock'sprice will be up at least30 percentafter the next 1000time periodsif u : 1.0\2, d : 0.990,andp : .52. 30. An image is partitioned into 2 regions--one white and the other black. A rcading taken from a mndomly chosenpoint in the white section will give a reading that is nomally distributed with trr : 4 ando2 : 4, whereasone taken from a mndomly chosenpoint in the black region will have a normally distributed reading with parameters(6, 9). A point is randomly chosenon the image and has a reading of 5. If the fraction of the image that is black is o, for what value of d would the Fobability of making an eflor be the same whether one concludedthe point was in the black region or in the white region? 31. (a) A fire stationis to be locatedalonga road of lengthA, A < a. If fires will occur at points uniformly chosenon (0, A), where should the station be located so as to rninimize the expecteddistanceftom the nre? That is, choosea so as to minimizeEIX - all whenX is uniformly distributedover (0, A). (b) Now supposethat the road is of infinite length-stetching ftom point 0 outward to oo. If the distance of a fire from point 0 is exponeltially distributed with rate ;", where should the fire station now be located? That is, we want to minimize E[lX - al], whereX is now exponentialwith rate.]". 32. The time (in hoursI requiredto repaira rnachineis an exponentiallydistributedrandom I : {. What is variablewith Darameter (a) the probability that a rcpair time exceeds2 hou$; (b) the conditional probability that a rePair takes at least 10 hours, given that its dulation exceedst hours? 33, The number of years a radio functions is exponentially distributed with parameter)" : probability that it will be working after an {. If Jonesbuys a usedradio, what is the additional8 years? 34. Jonesfigures that the total number of thousandsof miles that an auto can be driven beforc it would needto be junked is an exponential rundom variable with parameter S Smith has a usedcar that he clairnshasbeendriven only 10,000miles. If Jonespurchasesthe car, what is the probability that she would get at least 20,000 additional miles out of it? Repeatunder the assumptionthat the lifetime mileage of the car is not exponentially distdbuted but nther is (in thousandsof miles) uniformly distributed over (0, 40). 35. The lung cancerhazardmte ofa /-year-oldmale smoker,l(l), is suchthat ^\t):.o27 -.00025u 4ot2 tz4o Assuming that a 4O-year-oldmale smoker survives all other hazards,what is the probabitty that he survives to (a) age 50 and (b) age 60 without contacting lung cancer? 36. Supposethat the life distributionof an item hashazardmte function)'(/) : t3,t > O. What is the probability that Problems' 313 'E b v wherep is theirjoint probabilitymassfunction.Also, if X andf arejointly continuous with joint d€nsityfunction /, then the conditionalprobability den"sity function of X giventhat I = y is givenby IxvQli: f \:4 f v \.Y) vafuesX111.5Xot 1 X1,1of a setof independent andidenticay l!::L*:o ,,.. ! orstnbutedrandomvariablesare called the order statisticsof that set. If the random variablesarecontinuouswith densityfunction thenthejoint densityfunction /, of the orderstatisticsis J@t" ' x " ) : n t f ( x r )" ' f ( x , ) xt=x2=..'=xh The randomvariablesX1, . . . , Xn are calledexchangeable if thejoint distributionof X ir, . . ., X i, is the samefor everypermutation ir, . . ., i,, of 1, . . ., n _ PROBLEMS 1. Twofairdicearerolled.Findthejoint probabilitymassfunctionof X andf when (a) X is thelargestvalueobtained on anydieandy is thesumof thevatues: (b) X is thevalueonthefimtdieandy is thelargerof thetwo values; (c) X is thesmallest andy is thelargestvalueobtained onthedice. 2. Suppose that3 ballsarechosenwithoutreplacement from anum conslstrng of 5 white and8 redballs.LetX; equalI if thelth ballselected is white,andletit equal0 otherwise. Givethejoint prcbabiliry masstunctionof (a) X1,X2; (b) Xt, Xz, Xy 3. In-Problem2, suppose thatthe whireballsarenumbered, andler fi equalI if thelth whiteballis selected and0 otherwise. Findrhejointprobability massfunctionof (a) Yt, Y2: (b) Yt, Yz,Yz. 4. RepeatProblem2 whenthe ban selectedis replacedin the um beforcthe next serection. 5. RepeatProblem3a whentheba[ selectedis replacedin the um beforethe next selection. 6. A bin of5 tansistorsis known to contain2 that are defective.The transistors are to be tested,oneat a time, until the defectiveonesareidentified.Denoteby N1 the number of testsmadeuntil the first defectiveis identifledand by N2 the numbeiof additional tests until the seconddefectiveis identified;find thejoint probabilitymassfunctionof Nr and 7. Considera sequenceof independentBemoulli trials, each of which is a success with probabilityp. Let Xl be the numberof failuresprecedingthe flrct success, and tet X2be the numberof failuresbetweenthe fiIst two successes. Find the joint massfunctionof Xl andX2, 8. Thejoint probabilitydensityfunctionof X and y is sivenbv f ( x , y ) : c 1 y 2- y z p - t (a) Find c. (b) FindLhemarginal densiries of X and f. (c.) Find E[X]. -)<xs),0<-y<oo 314 Chapter6 JointlyDistributedRandomVariables 9. The joint probability density function of X and y is given by . r ( . r , v i)Q : '.+) o < x <r ,o . v . 2 (a) Verify that this is indeedajoint densityfunction. (b) Computethe densityfunctionof X. (c) Find P{X > YJ. (d) FindP{r r;X.;1. (e) Find E[x]. (f) Find E[Y]. 10. The joint probabilitydensityfunctionof X and f is givenby ir") / ( . r . t y )= e 0.x<oo.0.y<oo Find(a) P{X < Y} and(b) PIX < al. 11. A televisionstoreowner figurcsthat 45 percentofthe customersenteringhis storewill purchasean ordinarytelevisionset, 15percentwill purchasea plasmatelevisionset,and 40 percentwill just be browsing.If 5 custome$enterhis storc on a given day,what is the probabilitythathe will sell exactly2 ordinarysetsand 1 plasmaseton that day? 12. The number of people that enter a drugstorein a given hour i$a Poissonrudom vadable with pammeter)" : 10. Computethe conditionalprobabilitythat at most 3 men entered the drugstore,given that 10 women enteredin that hour. What assumptionshaveyou made? 13. A man and a woman agreeto meet at a certainlocation about 12:30P.M. If the man arrivesat a time uniformly distributedbetween12:15 and 72.45 and if the woman independentlyarrivesat a time uniformly distributedbetween12:00and I PM., lind the Fobability that the flISt to arrive waits no longer than 5 minutes. What is the probability that the man anives flIst? L4. An ambulancetravelsbackand forth, at a constantspeed,alonga rcad of lengthl. At a certainmomentof time an accidentoccursat a point uniformly distributedon the road. [That is, its distancefrom one of the fixed ends of the road is uniformly distributed over (0, t).1 Assumingthat the ambulance'slocation at the momentof the accidentis also uniformly dist buted,compute,assumingindependence, the distributionofits distance ftom the accident. 15. The randomvector(X, y) is saidto be unifondy distributedovera regionR in the plane if, for someconstantc, itsjoint densityis - c r, ,' ', r, ',-, ll 0 i f t . r . Y re f i otherwise (a) Showthat 1/c: areaofregion R. Supposethat (X, Y) is unifornly distributedover the squarecenteredat (0, 0), whose sidesarc oflength 2. (b) Show that X and f are independent,with eachbeing distdbutedunifbrmly over (-1. 1). (c) What is the probability that (X, y) lies in the circle of mdius 1 centeredat the origin?That is, find P{X2 + Y2 < \1. 16, Supposethat r points are independentlychosenat mndomon the perimeterof a circle. and we want the probabilitythat they all lie in somesemicircle.(That is, we want the probabilitythat therc is a line passingthroughthe centerof the circle suchthat all the pointsareon one sideof that line.) Problems315 Let Pt,. . ., Pz denotethe,' points. Let A denotethe eventthat all the points are contained in some semjcircle, and let A; be the event that all the points lie in the semicircle beginningat the point P; andgoing clockwisefor 180', i : l, . . ,n. (at ExpressA in lermsofthe Ai. (b) Are the A; mutually exclusive? (c) Find P(A). 17. Threepoints X l, X2, X3 are selectedat randomon a line Z. Wlat is the probabilitythat X2 lies betweenXt and X3? lE. Two points a{e selectedrandoniy on a line of length I so as to be on opposite sides of the midpoirt of the line, [In other words, the two points X and Y arc independent random variables such that X is uniformly distributed over (0, Ll2) and Y is uniformly distributed over (L/2, Z).1 Find the probability that the distancebetweenthe two points is geater than,L/3. 19. Showthat /(t, y) : l /x,0 < y < )c < I is ajoint densityfunction. Assumingthat / is thejoint densityfunctionof X, 1, find (a) the marginal density of Y; (b) the maryinal density of X; (c) E[x]: (d) Etvl. 20. The joint demity of X and y is givenby /(,, r) : {;'-''*" Are X and ts independent?What if /(t, ,rtx. ),= .r>0, y>0 otherwise y) were given by 12 0<x<y.0<)< l0 otherwise L 21. Let f6,y):24xy o5t:1' o5y:1' o5x*Yl1 and 1etit equal 0 otherwise. (a) Showthat /(r, )) is ajoint probabilitydensityfunction. (b) Find Elxl. (c) Find ElYl. 22. The joirt density function of x and y is y < I : l,+v 0 < x < l, 0 < "f(,r,r) l 0 otherwise 316 Chapter6 JointlyDistributedRandomVariables (a) Are X and Y independent? (b) Find the densityfunctionof X. (c)FindPIX+Y < 1]. 23. The randomvariablesX and I havejoint densityfunction. -f(r,y):12ry(1 "t) 0<r<1, 0<)<l and equalto 0 otherwise. (a) Are X and I/ independent? (b) Find E[x]. (c) Find E[Y]. (d) Find Var(x). (e) Find Var(Y). 24. Considerindependenttrials each of which resultsin outcomei, i : 0, l, ..., i with t 25, 26. 27. 28. 29. probabilityh , D p; ,:' | . Let ly' denotethe numberof trials neededto obtain an outi:0 comethat is not equalto 0, andlet X be that outcome. (a) FindP{N:nl,nZl. (b) FindP{x: j), j =r,...,k. ( c ) S h o wt h a tP { N : n , X = j \ = P { N : n \ P l X : j J . (d) Is it intuitiveto you that N is independent of X? (e) Is it intuitiveto you that X is independent of N? Supposethat 106peopleanive at a servicestationat timesthat areindependent mndom variabl€s,eachof which is uniformly distributedover (0, 106).Let ly' denotethe number that arive in the nrst hour.Find an approximationfor P{N : il. Supposethat A, B, C, arc independentrandom variables,each being uniformly distdbutedover (0, l). (a) What is thejoint cumulativedistuibutionfunctionof .A,B, C? (b) What is the probabilitythat all of the rootsofthe equationAx2 + Bx + C : 0 are real? If X is uniformly distributedover(0, l) and Y is exponentiallydistributedwith parameter ),: l, findthedistribution of(a) Z: X+Y afi (b) Z: X/l'. AssumeindependenceIf X1 andX2 are independentexponentialnndom vadableswith respectiveparameren 11 and.l"2,find the distributionof Z : Xt/Xz. Also computeP{X1 < X2l. When a current 1 (measuredin amperes)flows through a rcsistanceR (measuredin ohms),the power generatedis givenby Iry': 12R (measuredin watts). Supposethat / and R are independent rundomvariableswith densities -6.111 fi(.r) .fp(.:t)= 2x \) 0S,r<1 0<x<l Determine the density function of l/. 30. The expectednumberoftypogmphicalerrorson a pageofa certainmagazineis .2. What is theprobabilitythatanarticleof 10pagescontains(a) 0, and(b) 2 or moretypog.aphical errors?Explainyour reasoning! 31. The monthly worldwide avercgenumberof airplanecrashesof comnercial airlinesis 2.2. What is the probabilitythat therewill be (a) morethan2 suchaccidentsin the next month; (b) morethan4 suchaccidentsin the next 2 months; Problems 317 (c) more than 5 such accidentsin the next 3 months? Explail youl reasoning ! The gross weekly salesat a certain restaurantis a normal random vadable with mean $2200 and standarddeviation $230. What is the prcbability that (a) the total gross salesover the next 2 weeks exceeds$5000; (b) weekly salesexceed$2000 in at least 2 of the next 3 weeks? Wlat independenceassumptionshave you made? .'-t. Jill's bowling scoresare approximately normally distributed with mean 170 and standard deviation 20, while Jack's scoresaxeapproximately normally distributed with mean 160 and standarddeviation 15. If Jack and Jill eachbowl one game,then assumingthat their scoresare independentrandom variables, apprcximate the probability that (a) Jack'sscoreis higher; '' (b) the total of their scoresis above350. 34. Accordingto the U.S. NationalCenterfor Health Statistics,25.2 percentof malgsand 23.6 percent of females never eat breakfast. Supposethat nndom samplesof 200 meII and 200 women arebhosen.Approximate the probability that (a) at least110of these400 peoplenevereatbreakfasti (b) the number of the women who never eat breakfastis at least as large asthe number of the men who never eat breakfast. 35. In Problem 2, calculate the conditional probability massfunction of X1 given that (a) Xz = 1; (b) x2 = 0. 36. In Problem 4, calculate the conditional probability massfunction of Xt given that (a) X2 = l; (b) x2 : 0. 37. In Problem 3, calculatethe conditional probability massfunction of yl giYenthat (a) Y2: ll (b) v2:0. 38. In Problem 5, calculate the conditional probability massfunction of yl given that (7) Y2=l; (b) v2:0. 39. Choosea number X at randomfrom the set of numbers{1,2,3,4,51. Now choosea numberat randomfrom the subsetno largerthan X, that is, from {1,..., X}. Call this secondnumberI. (a) Find the joint massfunction of X and y. (t) Findtheconditional massfunctionofx giventhatf = i. Doitfori :1,2,3,4,5. (c) Are X and l independent?Why? 40. Two dice are rolled. Let X and Y denote, respectively,the largest and smallest values obtained.Cornputethe conditionalmassfunctionof I givenX : i, for i : 1,2, .. .,6. Are X and Y independent?Why? 4L. The joint Fobability massfunction of X and y is given by p(l,l): * P ( r , 2 ) :i p ( 2 , \ ) :i p Q , 2 ) :+ (a) Computethe conditionalmassfunctionof X given Y : i,i : 1,2. (b) Are X and Y independent? (c)compute P{xv <31,Plx +Y >21,PlxlY > ll. 3 1 8 C h a p t e6r JointlyDistributedRandomVariables 42. The joint densityfunctionof X and Y is givenby f(x,y):xe r()+r) ;r>0, ),>o (a) Find the conditionaldensityof X, given f : )], andthat of I, givenX : .r. (b) Find the densityfunctionof Z = Xy. 43. Thejoint densityof X and Y is f ( : r ,D : c ( x 2- ! 2 ) e - ' o<ir<c!, -r:y:.r Find the conditionaldistributionof Y, given X : -r. 44, An insurancecompanysupposesthat eachpersonhas an accidentpammeterand that the yearly nurnberof accidentsof someonewhose accidentparameteris .1"is poisson dist butedwith mean),. They alsosupposethat the pammetervalueof a newly insured personcan be assumedto be the valueof a gammarandomvariablewith pammeters.t and a. If a newly insuredpersonhas n accidentsin her flrst year,find the conditional densityofher accidentparameterAlso, determinethe expectednumberofaccidentsthat shewill havein the following year. 45. lf X1, X2, X3 are independentrandomvadablesthat are uniformly distributedover (0. l), computethe probability that the largestof the three is greaterthan the sum of the othertwo. 46. A complexmachineis ableto operateeffectivelyaslong as at least3 ofits 5 motorsare functioning. If eachmotor independentlyfunctionsfor a randomamountof time wirh densityfunction/(r) = xe-' , x > 0, computethe densityfunctionofthe lengthof time that the machinefulctions. 47. If 3 fucks break down at points randomlydistriburedon a roadtf length t, iind the probabilitythat no 2 of the tucks arc wirhin a distanced of eachotherwhend : L/2. 48. Considera sampleof size5 from a uniform distributionover (0, 1). Computethe probability that the medianis in the interval ( {, ol). 49. If Xt. X2, X3, X4, X5 are independentand identically distributedexponentialrandom variables with the parameter)", compute (a) P{min(Xr,...,X5) : rr}; ( b ) P [ m a x ( X 1 , . . .X, 5 ) : a ] . 50. Derive the distributionof the range of a sampleof size 2 from a distributionhaving densityfunction /(r) = 2r,0 < r < 1. 51. Let X and y denotethe coordinatesof a point unifondy chosenin the circle of radiusI centeredat the origin. That is, theirjoint densityis I x2+!2=1 7f Find_thejoint densityfunction of the polar coodinates R : (X2 + y\1/2 and@ an tYlx. 52. If X and I areindependentrandomvariablesboth uniformly distributedover (0, l), find t h e j o i n dr e n s i t fyu n c r i o on f R - , , - x , - Y ' - O - - t a n - t y / x . { ? If U is uniform on (0, 2z) and Z, independentof U, is exponentialwith rate 1, show dircctly (withoutusingthe resultsof Example7b) thar X and Y definedby X : J22co:u Y : xEZ sinU areindependent standardnormalrandomvariables. Theoretical Exercises 319 54. X and I havejoint densityfunchon ytr. tt: 9ter and that I is Poisson twly insured Dalamelen.r ) conditional rcidentsthar ftedover (0. I sum of the i motors are f time with ngthof time l,, find the d < L/2. e the probatial random fon having ofradiusI (0, l), find Cel, shou )zrz .r> r. ),: r (a) Computerhejoint deNity functionof U = Xy,V = X/y. (b) Whar arc the marginal densities? 55. If X arld f are independentand identically distributed uniform random variables on (0, l), computethejoint densityof (a) U=X+y,V:X/y; (b) U : X,V : X/Y; ( c )U = X + Y , V : X / ( x + y ) . 56. RepeatProblem 55 when X and I are independentexponential mndom variables, each with parameter), = l. 57, tf X1.andX" are independent exponential randomvariables each """' havrngparameter )., findthejointdersig,functionof I/1: Xt!Xzandyz:ei,. _58. If,X, y, and arc independent_random variableshaving identical density functions -Z e', 0 < r < oo, derivethejoint distributionof U": i i i, v = X + z, w = dfi) ; 59. In Example8b,1etY1a1 : na1. fy,. Showtharyr..... y*,yr+tare exchangeable. Note that fft+l is the number of billi one must observe to obtain a special ball if one consldersthe ballsin their reverseorderof withdrawal. 60. Consideran.urncontaining/, balls,numberedl, .. ., z, and supposethat f of them are randomllwirhdrawn.Ler Xi equalI if ball numbered i i, re.o'u!aunOlerir be 0 orher_ w l s e .) n o w l h a lX t . . . . . & a { ee x c h a n s e a b l e . THEORETICAL EXERCTSES 1. VeriryEquation (1.2). 2. Supposethat the numberof eventsthat occurin a given time periodis a poissorrandom variable with parameter,,. If each event is classified as a type I event with probability pi,i = 1,...,n,Dp; = 1, independently of otherevents,showthatthe numbersof t)4'e i eventsthat occur,i : 1,...,n, are independentpoisson mndom variableswith respecnveparameteN ),pi,i : 1,..., n, 3. Suggesta procedurefor using Buffon,s needleproblem to estimatez. Suryrisingly enough,this wasoncea commonmethodof evaluatinsz. 4. SolveBuffon s needleproblemwhenI > D. a N s w e n :1 1 ( l - s in e | + 2 e/ n . w h ere d i s s u c hr h a cr o s 0: D/ L. 5. If X and f are independentcontinuous positive random variables, expressthe density functior of (a) z = X /Y afi (b) z = Xr in termsof the a"nrity iun"tion, or x una y. Evaluatetheseexpressions in the specialcasewhere X and y"ale mtn exponential random va.riables. 6. Showanalytically(by induction)that Xr * . . . * Xn hasa negative binomiatdistribution whenthe X;, I : 7, . . -, n a.reindependent artdidenticattydiitributeJgeornetricrandom variables..Also, give a secold argumentthat yedfies the ;receding wiitout any need tor computahons. 7. (a) If X has a gammadistributionwith paramete$(r,.i.), what is the distributionof cX,c>0?