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results by means of a (b) Evaluate F (x). (c) What is the probability that a random particle 3.31 from Based on extensivefuel testing, determined by the manufactured exceedsit4 is micrometers? 3.34 Magnetron 4 dimes and 2 nickels, the manufacturer of a washing machine that the time assembly line. A without replacement. Y 3.30 (in years) before aofmajor repair is required is char- assess quality of Measurements scientific systems are always n for the totalEX T 1: of Measurements the subject of variation, someare more than others. There some more scientific systems always subject to variation, than surement is subj acterized to by the probability density function ty distribution graphiaremany manystructures structures for measurement error, and statisothers. There are for measurement error, and statisticians spend a great deal of probability t ! 1 −y/4 94 Chapter 3 the Random Variab . spendSuppose a greatthe deal of time modeling these errors. time modelingticians these errors. measurement error X of a certain physical quantity is e , y ≥ 0, ification is 0.99. f (y) = 4 the measurement error X of a certain physical decided by theSuppose density function elsewhere. lengths of 5 rand black balls and 2 green quantity is decided 0, by the density function ession, each ball being 3.31 Based on Magnetron testing, it is determined by 3.34 ! extensive (a) Show that tht 2 k(3 − x ), −1 ≤ x ≤ 1, ext draw is made. Find (a)the Critics would certainly consider the product a barline.ofA5 sa manufacturer of a washing machine that the time assembly f (x) = ber out t r the number of green Y gain 0, a major elsewhere. if it is before unlikely to require a major repair (in years) repair is required is before char- assess quality of th by the follow sixth Comment on this by determining surement is subjec acterized by year. the probability density function (a)the Determine k that renders f (x) a valid density funca) Determine k(Y that>renders density function. Sol:the 3/16probability tha Ption. 6). f(x)!a valid hours of an important 1 −y/4 e , y ≥ 0, b) Find the probability that a random error in measurement is less than 1/2. Sol: 99/128 is f (y) = ification 0.99. A 4 sed in a manufactured(b) isfthe probability a major occurs (b)What Find the probability that that a random errorrepair in mea(y) = 94 Chapter 3 Random Variables and P c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., |x|) 0, 1/2. elsewhere. lengths of 5 random nction is less than insurement the exceeds 0.8. Whatfirst is theyear? probability that this occurs? Sol: 0.164 (a) Show (c) For this particular measurement, it is undesirable for ythat = 0,the 1, 2p 00), x ≥ 0, (a) 3.31 Critics would certainly consider product 3.34 Magnetron tubes are Based on extensive testing, it isthe determined bya barif the magnitude of the error (i.e., |x|) exceeds 0.8. ber out of 5 tha EX 2: Based on extensive testing, it is determined by the manufacturer of a washing machine 3.32 the The budget for certain type assembly line. A sampling p manufacturer of aof washing machine thatarepair the time x < 0. gain ifisproportion itthe is unlikely tothe require aoccurs? major before (b) Suppose rand What probability that this by of thethefollowin that the time (inYyears) before a major repairisisrepair required isto characterized by assess the probability quality lengths (in years) before a major is required is charofYindustrial company that allotted environmental the sixth year. Comment on this by determining 3 are out density function surement is and subject to uncer acterized bycontrol the probability density function and pollution is coming under scrutiny. A data P (Y > 6). ther toasuppo the probability that random ! 1 −y/4 collection project determines that the distribution of e , y ≥ 0, f (y) = is ification is 0.99. A sampling 4 probability (b) What isfthe probability that a major repair occurs (y) = y! these in proportions is given by 0, elsewhere. lengths of 5 random tubes a the first year? cations. (a) Show that the probabilit ! certainly a) Critics would consider the product a bargain if it is unlikely 4consider (a) certainly Critics would the product a bar- to require a major = 0,meet 1, 2,len 3 5(1 − y) , 0 ≤ y ≤ 1, ber 0.2231 outfor of 5ythat repair before the year. Comment on this by determining P(Y >6). ifproportion it= is unlikely to require a major before fsixth (y) 3.35 Suppose it 3.32 gain The of the budget forrepair a certain typeSol: by (b) the following Supposediscrete rando b) What is the probability a major repair occurs in the first year? Sol: 0.2212 0, elsewhere. the sixththat year. Comment on this by determining torical data that of industrial company that is allotted to environmental and 3 are5!outsi P (Y > 6). a specific interse and pollution control coming under scrutiny. A data f (y) (0 EX 3: The (a) proportion of that the is budget for aiscertain industrial that is allotted to= to support ther Verify theprobability above isthat atype valid density function. (b) What the a of major repaircompany occurs y!(5 − y)! collection project determines that the distribution of is characterized environmental and pollution control under scrutiny. A data collection project in the first year?is comingthat probability is 0 (b) What is the probability a company chosen at function: these proportions is given by determines that the distribution of these proportions is given by 0, 1, 2, 3, 4, 5. random expends less than 10% of its budget on en-for y =cations. ! 3.32 The proportion of the budget for a certain type (b) Suppose random selectio 4 vironmental and pollution 5(1 − y)is, allotted 0controls? ≤ yto≤environmental 1, of industrial company that and3.35 3 are outside specific f (y) = Suppose f (x) =itei− and pollution under A data 0, is comingthat elsewhere. (c) What is the control probability ascrutiny. company selectedthertorical to support to refu dataorthat X collection project determines that 50% the distribution of probability is 0.99 that a at random spends more than of its budget on a) Verify (a) thatthese the above is a valid density function. Sol: ? cations. a specific intersect proportions isabove given by Verify that theand is a valid density function. Find the pro environmental pollution controls? b) What is the probability that ! a company chosen at random expends less than 10% of its is(a) characterized by 4 5(1 − y) controls? , that 0 ≤ y a≤company 1, time period, What isf (y) the= probability chosen at function: budget(b) on environmental and pollution Sol:3.35 0.4095 Suppose it is known f 0, less than elsewhere. expends 10% of itsdata budget on enintersection. c) What is therandom probability that a company selected at random spends more than 50% of its torical data that X, the num 3.33 Suppose a certain type of small processing vironmental and pollution controls? budget on environmental and pollution controls? Sol: 0.031245 a specific intersection during the pro −6 firm is specialized that issome difficulty (a)so Verify that the above a validhave density function.making (b) Find fby (x)the = efollo is characterized (c) What is the probability that a company selected a profit in type their first year ofthat operation. probabilWhat is the probability a company The at function: EX 4: Suppose a (b) certain of small data processing firm is sochosen specialized that some have at random spends more than 50% of its budget on random expends less than 10% of its budget on enity density function thatofcharacterizes the proportion 3.36 difficulty making a profit in their first year operation. The probability density function that On xa labo vironmental and pollution controls? (a) Find−6the 6 proba environmental and pollution controls? Y the that makeYathat profit isa profit givenis by characterizes proportion make given by f (x) = e the , for x working, den (c) What is the probability that a company selected time period, m x! ! X,intersection. is 4 3than 50% of its budget on at random ky spends more (1 − y) , 0 ≤ y ≤ 1, 3.33 fenvironmental Suppose a certain type of small data processing (a) Find the probability tha and pollution controls? (y) = (b)period, Find more the proba firm is so specialized that someelsewhere. have difficulty making time 0, than 8 intersection. a profit in their first year of operation. The probabilf (x) = a certain of asmall processing a) What is the3.33 value Suppose of k that renders thetype above valid data density function? Sol: 280 (a)ityWhat the valuethat of ofsome kcharacterizes that the above (b) aFind the probability tha firm is is sothat specialized that haverenders difficulty making density function proportion 3.36 On a labora b) Find the probability at most 50% the firms make athe profit in the first year. a profit in their first year of operation. The probabilvalid density function? Y that make a profit is given by Sol: 0.3633 the densit ity density function that characterizes the proportion 3.36 working, On (a)aislaboratory Calculateassign P( ! c) Find(b) the Find probability that at least 80% of the firms make a profit in the first year. the probability that at most 50% of the firms X, 4 is given3 by Y that make a profit working, the density function ky (1 − y) , 0 ≤ y ≤ 1, Sol: 0.0563 make fa(y) profit year. = !in 4the first X, is (b) What is the! ky (1 − y)3 , 0 ≤ y ≤ 1, 0, elsewhere. f (y) = that X (c) Find the probability that at least 80% of the firms (c) Given !f (x) 0, elsewhere. =x), 2(1 − X=will be les make a profit the of first year.renders the above a f (x) (a) What is the in value k that 0, (a) What is the value of k that renders the above a valid density function? valid density function? (a) Calculate P (X (b) (b) Find the thatatatmost most 50% of firms the firms (a) Calculate P (X ≤ 1/3). Find theprobability probability that 50% of the (b) isWhat is the prt make a aprofit firstyear. year. (b) What the probability make profit in in the the first 3.4 Joint Probability Distributions f (x) = , for thetobaccos. weight of her Thethe propor- (b) Find the accounts probability first is busy more tobacco forthat overthe half theline blend. x! stic in a blend are random than 75% of the time. Findamount the marginal density function for the propor3.47(b)The of kerosene, in thousands of liters,(a) Determine the probabili function (X = Turkish and tion of the domestic tobacco. eight of the toffees in a tank 4, 5, and 6. at the the number beginningphone of any day is a by random 3.65 Let received a (c) Find the probabilityofthat the calls proportion of Turkgram if it is known amount Y fromduring whichaa5-minute randominterval amountbeXa israndom sold dur-(b) Graph the probability m switchboard x, yweight. ≤ 1, x + y ≤ 1, ish tobacco is less than 1/8 if it is known that the he variable X with probability function ing that day. Suppose that the tank is not resupplied ues of x. where. blend contains domestic tobacco. EX 5: Let X denote the diameter of an3/4 armored electric cable and Y denote the (c) diameter of the Determine the cumulativ during the the daycable. so−2that xand ≤ Yy,areand assume thatrange thebetween x ceramic mold that makes Both X scaled so that they and of X. e 2 at a given the Turkish these 0values thsin of life, box in1.years, f (x) = , for x = 0, 1, 2, . . . . joint density function of these variables is company offers its policyholders a Suppose that X3.62 and YAn haveinsurance the joint ver half the blend. x! density system. If the joint number of different premium payment options. For a ! ity function for the propor2, 0 < x ≤letyXX<equals 1,the 0,number (a) Determine the probability that 1, 2, 3,of 3.66 Consider the random sbacco. randomly selected policyholder, be / / joint density function 4, f 5,(x, andy)6.= successive months between payments. The cumulative 0, elsewhere. hat the proportion of Turk- distribution function ! of Xmass is function for these valy1/8 > if0,it is known that the (b) Graph the probability x + y, ⎧ f (x, y) = uesand of Yx.are a) Determine if X independent. ⎪ 0, if x < 1, (a) Determine if X and Y are independent. mestic ⎪ 0, ere, tobacco. Chapter 3 Random Variables ⎪ and Probability Distributions Sol: 1/3 ⎪ b) Find P(¼ (c) < XDetermine < ½ | Y = ¾the ) (!cumulative ⎪ 0.4, if 1distribution ≤Probability) x < 3, function for ⎨Conditional these values of X. (a) Find the marginal distri F (x) = 0.6, if 3 ≤ x < 5, ny offers its policyholders a ⎪ ⎪ um payment options. For a 3.53 Given the joint ⎪0.8,density (b) Find P (X > 0.5, Y > 0. if 5 ≤ function x < 7, = 3/4). Consider the⎪ ⎪ EX number 6: Givenofthe 3.66 joint density function ⎩random variables X and Y with lder, let X be the 1.0, if x ≥ 7. joint density function ! 6−x−y payments. The cumulative / 4,/ 3.67 An industrial proces , 0 <mass x <function 2, 2 < yof < ! 9, 8 (a)f What is the probability X? is find (x, y) = can be classified as either x + y, 0 ≤ x, y ≤ 1, f (x, y)P0, = (b) Compute (4 < X ≤ elsewhere, 7). The probability that an ite f x < 1, 0, elsewhere. find P( 1 < Y < 3 | X = 1) (! Conditional Probability). 5/8 is conducted experiment in f 1 ≤ x < 3, 108 Chapter 3Sol:Random Variables find P (1 < Y < 3 | X = 1). randomly from the process. 3.63 Two components of X a missile (a) Find theelectronic marginal distributions and Y . system f 3 ≤ x < 5, the number work in harmony for the success of the total system. EX 7: Two electronic components of a missile system work in harmony for thebesuccess of the of defectives (b) Find P (X > 0.5, Y > 0.5). fr 5of≤times x < 7,a certain nuthe probability mass Let XDetermine Y its denote lifethe in of the two comwhether two random variables of total system.(a) Let3.54 X and Yand denote the lifethe in hours ofhours the two components. The is joint density of time 3.74 The Z func in m Determine probability density function. f x ≥ 7. ponents. The joint density of X and Y is malfunction: 1, 2, or 3 X and Y is Exercise 3.49 are dependent or independent. trical supply system h An industrial manufactures (b) 3.67 Determine the probability that the lifeitems spanthat of such3.68 ! process Consider the followin the number −y(1+x) y denote mass function of X? of tion , x, y ≥ 0, ye can be classified as either defective or not defective. a component exceed 70 hours. function of the random vari f (x, y) =will n).emergency call. Their 3.55 Determine whether the two random variables of !1 − The probability that iselsewhere. defective is 0.1. An 0, an item ! 3x−y e given as experiment is are conducted in which 5 items are drawn Exercise 3.50 dependent or independent. 1< f (z)9= , 10 f (x, y) = a) Give the marginal density functions for both random variables. Sol: ? (a) Give the marginal density functions for both ran3.70 Pairs of pants are being produced by a particux 0, ponents of a missile system randomly from the process. Let the random variable X 0, elsew dom variables. b) system. Whatlar is the probability that the lives of both components will exceed 2ofhours? outlet facility. The pants are checked by a group be the number of defectives in this sample of 5. What 2 success of the3 total 3.56 The joint density function of the random vari6 is the mass function X? (b) What isThe the probability thatofthe livesof of pants both com10ables workers. inspect pairs taken(a) Sol: (a) What is the densi prob fe in hours 0.10 of the two com0.05 Find1/(3e the )marginal Xprobability and Y workers is ponents will exceed 2 hours? of X and Y0.35 is randomly from the production line. Each inspector is(b) Arewithin 0.10 X and a Y 20-minut independe ! the 3.68 Consider following joint probability density EX 8: Consider the following joint probability density function of the random variables X and x) assigned a number from 1 through 10. A buyer selects 0.20 0.10 6x, 0 < x < 1, 0 < y < 1 − x, is 2). the prob , x, y ≥ 0, FindWhat P (X > function random variables X and Y :service lines. (c) (b) Y: f (x, y)of =thefor 3.64 service facility operates with two a pair of A pants purchase. Let the random variable within 10 minutes elsewhere. elsewhere. ibution of X. !0, Onthe a randomly selected day, let X 1be the proportion of 3x−y X be inspector number. , 1 < x < 3, < y < 2, 9 line is in use whereas Y is the pro3.69 The life span in hou that y)the = first ibution of Y .for both ran- timef (x, sity functions 0, elsewhere. (a)(a) Give a reasonable probability mass for X.nent3.75 is a random variable w Showofthat and Y second are not independent. portion timeX that the line is infunction use. Suppose A chemical syst function that the joint probability density function for (X, Y ) is (b) Plot the cumulative for X. reaction has two impo a) Find the(b) marginal functions of=X 0.5). and Y. function Sol: ? y that the lives of both comP (Xmarginal > 0.3 |density Ydistribution (a)Find Finddensity the functions of X and Y . ! ! x 2 2 − 50 in a blend. The 3 ours? b) Are X and(b) Y independent? Sol: ? 1 − ejoint (x + y ), 0 ≤ x, y ≤ 1, , Are X and Y independent? 2 have the following joint F (x)X =1 and X2 o f (x, y) = portions c) Find P( X > 2). Sol: 2/3 3.71 The shelf life of a product is a random variable 0, 0, 3.57 LetP X, Y ,2). and Z have elsewhere. the joint probability den(c) Find (X > by erates with two service lines. that is related to consumer acceptance. It turns out sity of function x let X be the EX 9: The shelf ashelf product variable that is related to y, proportion of ! thatlife the lifeisspan Ya random inindays of ofa certain typecompoofconsumer bakery acceptance. It The life hours an electrical 2 n use4 whereas turns Y is out the that pro-the3.69 ! shelf life Y in days of a certain type of bakery product has a density function 2 function product has a density f (x1 , x2 ) = nent is a random variable with cumulative distribution kxy z, 0 < x, y < 1, 0 < z < 2, cond line is in use. Suppose 0.15 0 f (x, y, z) = ! ensity 0,1 −y/2 elsewhere. 0.30function for (X, Y ) is function e ,x 0 ≤ y < ∞, (a) Give the marginal f (y) = !21 − e− 50 ), 0.15 0 ≤ x, y ≤ 1, , x > 0, 0, elsewhere. F (x) = (a) Find k. (b) Give the marginal 0, eleswhere. ion elsewhere. of X. 1 (b) Find P (Xof<the , Y > 1 , 1 < Z < 2). (c) What is the prob What fraction 4 loaves2 of this product stocked toWhat fractionday of the loaves of this product stocked today would you expect to be sellable 3 daysproduce the tions would you expect to be sellable 3 days from now? from now? without replacement 3.58 Determine whether the two random variables of (d) Give the condition Passenger congestion is aorservice problem in airExercise 3.43 are dependent independent. queens, and kings) of 3.72 cards. Let X be the ports. Trains are installed within the airport to reduce 3.76 Consider the si congestion. With the usethe of the the variables time X inof Determine whether twotrain, random Y the number of jacks. the3.59 But suppose the joint minutes that it takes to travel or from the main terminal Exercise 3.44 are dependent independent. tions is given by to a particular concourse has density function ! ution of X and Y ; 3.60 The joint probability density function of the ran6x !1 f (x1 , x2 ) = s the region given by dom variables X, Y10, ,and0 Z ≤ is x ≤ 10, 0, f (x) = 0, elsewhere. " 4xyz 2 (a) Give the marginal , 0 < x, y < 1, 0 < z < 3, 9 ion of Y . n? / / ⎩ 0, elsewhere. 117 que jewelry dealer is interthe average number of hours per year thatthe families random variable Y = 3X − 2, where X has density on pagefor 120which to find vari- Find necklace thethe probrun their vacuum cleaners. able of Exercise 4.7 onthat page function 127 , andX0.14, respectively, ! or a profit of $250, sell it for 4.14 Find the proportion 1 −x/4 who can be e X, of xindividuals >0 4 n, or sell it for aEX loss of $150. f (x) = expected to respond to a certain mail-order solicitation 10: Find the proportion X of individuals who can be expected to respond to a certain mail losses, what premium ion of X, the number of 0.1. Ignoring all other0, partial elsewhere. m with the following t? variable if X has the density function order solicitation if X has the density function a synthetic fabric in con- should the insurance company charge each year to re% variance an average profit of $500? of the random variable Y . given in Exercise 3.13for alizeFind the mean and esis to insure airplane −2 3 120 5his < x <X1,has the density 3X −, 2,0where on page to find the vari- random variable Y =2(x+2) 5 ompany estimates that a tof (x) = .3 X 0.2 0.5 4.7 on page function ble of Exercise 0, elsewhere. 4.44 Find the covariance of the random variables experts examine stacks of tiresX obability 2tion 3 0.002,4 a 50% loss 4.10 Two tire-quality of X. ! and Y of Exercise 3.39 page0 105. −x/4 onto a 25%0.05 loss with quality14 erating tire on a 3-pointSol: 8/15 , x >each 16 0.01probability and assign fa(x) = scale. Let X denote the rating given by expert A and m variable with the following ble X, representing 0, elsewhere. mperfections per 10 the me-num- 4.45 Find the covariance of theThe random variables Y denote the rating given by B. following tableX nes of softwareEX code, has proportion the and of 11: The people who respond to a certain mail order solicitation is a random Y of Exercise 3.49 on page 106. gives the X and Y . variable Y . Find thejoint meandistribution and variancefor of the random 2tribution: 3 5 variable X having the density function given in EX 10. Find the variance of X. 3of 0.2 0.5discrete of 4the 3ionRandom 5 Variables 6 ran119Sol: 4.46Find Find covariance ofy the random variables X 37/450 4.44 thethe covariance of the random variables X f (x, y) 1 2105.105. 3 25 of 0.4 on X. 0.3 0.04 and Exercise 3.44 page and Y Yof of Exercise 3.39 on on page 1 respond 0.10 to0.05 0.02 The proportion of people who a certain mail order solicitation is a random page 121, find EX the 12: variance of le X, representing the numx 2 0.10 0.35 0.05 What is the population mean of the times repair? 4.47 For the random variables X andvariables Ytowhose 4.45 Find function the covariance ofEX the10. random Xjoint , x = 0, 1, 2,variable 3. X having the density given in es of software code, has the and 2 3 3.49 0.03 Y of 106. 0.20 density function is+given in 0.10 Exercise 3.40 on page 105, Find theinExpected value of Exercise g(X) = 3X 4.on page Sol: 5.1 ,ibution: or mean weight, find the covariance. probabilities are 0.4, 0.3, 4.31 0.2, Consider Exercise 3.32 on page 94. and µ . Find µ X Y 5 3 power 6 at 0, 41, 2, or failures 4.46 Find the covariance of the random variables X (a) What isGiven the mean allo5 0.4 0.3 0.04 nswer in (b)? Explain 4.48 a random X,the withbudget standard deY of Exercise 3.44proportion on variable page 105.of ivision in any given year. Find and 4.11 The density function of coded measurements of threads of a fitting EX 13: The density function of coded measurements of the pitch diameter of ndom variable T reprecated to environmental and pollution control? viation σ , and a random variable Y = a + bX, show X age find variable the variance of the121, random X repretheWhat pitch threads ofcoefficient aafitting is 3.25this oins failures in Exercise on sub4.47 the variables X and Y ρis whose joint that For ifisdiameter b the < 0,random theofcorrelation −1, and (b) probability that company selected XY = ower striking % density function is given in Exercise 3.40 on page 105, if b > 0, ρ = 1. at randomXYwill have allocated to environmental dealt with an impor4 , 0 < x < 1, find the covariance. robabilities are 0.4, 0.3, 0.2, and π(1+x2 pollution a) proportion that exceeds the f (x) = control aracterized t in 0,a units 1, 2, or powertimes failures at head is 3by three 4.49 Consider the situation in Exercise 4.32 on page of $5000, on a new population 0, mean given in elsewhere. (a)? Given a random variable X, with of standard devision inexpected any givennumber year. Find 4.48 nd the 119. The distribution of the number imperfections variable X having the density 1, viation σ , and a random variable Y = a + bX, show X the random variable X repreexpected value10 of meters X Sol: ln4/π ofvalue synthetic failure is given by dsetwice. 4.12 on pageFind 117.theFind the Findper thebexpected of X. < 0, the 3.13 correlation coefficient ρXY = −1, and of wer failures striking this subwhere. 4.32thatInif Exercise on page 92, the distribution x 0 1 2 3 4 b > 0, ρof XY = 1. the if number imperfections per 10 meters of synthetic of the number of imperfections per 10 meters of synthetic fabric is woman is paidEX $314: if the shedistribution f (x) 0.41 0.16of $5000, 0.05 0.01 4.12 If a dealer’s profit,0.37 in units on a new respond toora cerfabric is given by given by 5npeople ifunits she who draws a king 4.49 thelooked situation in Exercise 4.32 of onvariable page of $5000, on aXnew automobile be upon asdeviation a random FindConsider thecan variance and standard the numon a random variable ofis52 playing If hav- 119. x 0 1 2 3 4 The distribution of the number of imperfections ariable X havingcards. the density X having the density function ber of imperfections. given in Exercise 4.14 on page per 10 meters of synthetic failure is given by oses. How much 4.12 on page 117.should Find the f(x) 0.41& 0.37 0.16 0.05 0.01 ffair? X. xprobability 0 2(1 1x),assignment, 94, the distribution of −function. 02< x <3 1, 4.50 For a laboratory if the 4equipmentSol: is 0.88 (a)expected Plot the a) Find the number of imperfections. f (x) = f (x) 0.41 0.37 0.16 0.05 0.01 outcome washing machine was function of the observed of hours, in units of 100 hours, 2 working, the density 0, elsewhere, people who respond atocerb) toFind E(X ). (b) Find expected number of imperfections,Sol: 1.62 ash is cleaner paid according X isthethe over a period of Find variance and standard deviation of the num- Sol: 0.8456 nuum is a random variable X havc) Find Var(X) E(X) = µ. through. Suppose the ariable X having the density ! per automobile. ber of imperfections. profit iven in Exercise 4.14 on pagefind the average 2 1/4, 1/4, 1/6, and 1/6, 2(1 − x), 0 < x < 1, 0, se the Find E(X f). X.4.13 on page 117. Find (c) (x) = nt receives $7, $9, $11, 0, assignment, otherwise. 4.50The Fordensity a laboratory if the equipment is where. 4.13 function of the continuous random EX 15: For a laboratory assignment, if the equipment is working, the density function of the P.M. and 5:00 P.M. on working, the density function of the observed outcome f hours, in units of 100 hours,variable X, the total number of hours, in units of 100 observed outcome endant’s earnrcise 4.14expected on page 117, find XXFind isis the variance and standard deviation of X. uum cleaner over a period ofhours, that a family runs a vacuum cleaner over a pe2 X) = 3X + 4. the density riable X having ! riod 4.51 of oneFor year, is given in 0Exercise 3.7 Yoninpage 92 2(1 − x),variables <x< the random X 1,and Exercise 4.13 on page 117. Find the (x) = as 3.39 onfpage 105, the correlation coefficient 0, determine otherwise. lar stock, ofa the person can varideviation random ⎧ between X and Y . 000 with probability n Exercise 4.17Find on page 118. and standard deviation x,Xstandard 0< x < 1, importance ⎨and the 0.3 variance ofisX.of ciseor4.14 on page 117, find the variance deviation of X. n, expected value, of a Find random variable special inSol: 1/18, 0.2357 robability 0.7. What is 2 X) = 3X + 4. f (x) = 4.52 Random variables X and Y follow a joint distri2 − x, 1 ≤ x < 2, because describes theFor probability is centered. By of Exerciseit4.21 on page where 118, 4.51 ⎩ distribution the random variables X and Y in Exercise bution EX 16: Random variables X and Y follow a joint distribution 2 0, elsewhere. wever, mean not give an adequate description of the shape of the ) = X the , where X isdoes a random the correlation coefficient ! deviation of the random vari- 3.39 on page 105, determine ity function given in Exercise on. We also need to characterize the variability in the distribution. In between X and Y . jewelry dealer inter2, 0 < x ≤ y < 1, Exercise 4.17 on is page 118. Find the average number of hours per year that families f (x, y) = which thehistograms prob1,lace weforhave the of twovacuum discrete probability distributions that 0, otherwise. run4.52 theirRandom cleaners. variables X and Y follow a joint distrid 0.14, respectively, that same mean, = page 2, but considerably in variability, f Exercise 4.21µ on 118, Determine thediffer correlation coefficient between X and Y or the dispersion bution e, in2minutes, for an airplane profit of $250, sell it for Determine the correlation between X be and = X , where X is a random bservations about the mean. X ofcoefficient individuals who can ! akeoff is4.14 a Y .Find the proportion sell it at fora acertain loss of $150. ty function given inairport Exercise 2, 0 < x ≤ y < 1, expected tofrespond (x, y) = to a certain mail-order solicitation 0, otherwise. if X has the density function Covariance of Random Variables , insure in minutes, for an airplane his airplane for keoff at a certain airport any estimates that a to-is a bility 0.002, a 50% loss 5% loss with probability % Determine the correlation 2(x+2) coefficient between X and , 0 < x < 1, 5 Y. f (x) = 0, elsewhere. 2 − x, 1 ≤ x < 2, − valuate E(2XY ⎩ 0, elsewhere. tribution shown in X Y) Table green die is tossed and Y the number that occurs when a certain 2-minute period day. ofThe distria red die is tossed. Find in thethe variance the joint random variable bution is % &% & o evaluate the mean of the random (a) 2X − Y ; 2 1 9 + 39X, where Y is equal to the (b) X +f3Y −y) 5. = (x, , (x+y) pendent randomannually. variables t hours expended 16 4 / / σY2 = 3, find the variance 4.67 If the joint density function of X and Y is given EX 17:such If the joint density function of X and Y is given by variable X is−defined that for xby= 0, 1, 2, . . . and y = 0, 1, 2, . . . . −2X + 4Y 3. $ 2 E(Y ), Var(X), and Var(Y ). (a) GiveExercises E(X), = 10 and E[(X − 2)2 ] = 6, (x + 2y), 0 < x < 1, 1 < y < 2, if X and Y are not inde- Review f (x, y) = Z 7 = X + Y , the sum of the two. Find (b) Consider 0, elsewhere, E(Z) and Var(Z). (b) Itfind is the of interest know something pro- Then do it not by usi expected3to value of g(X, Y ) = YX3 about + X 2 Y the . t are X and Y are independent random 2 independent random find the expected value of g(X,Y)= (X/Y )+X Y. 46/63 portion of Z = X + Y , the sum of the two proporthe first-order Taylor se he joint probability distribution 4.70 Consider Review Exercise 3.64 on page 107. Sol: nsities and tions. Find E(X + Y ). Also find E(XY ). 4.68 The power P in watts which is dissipated in an Comment! x There are lines. two service lines. The random variables X EX twoFind service The random variables X and Y are the proportions of time electric circuit with resistance R is known to be given x, y) 2 4 18: There are (c) Var(X), Var(Y ), and Cov(X, Y ).line 1 and line 2 and Y are the proportions of time that > 2, by P = I R, where I is current in amperes and R is a that line 1 and line 2 are in use, respectively. The joint probability density function4.75 for (X,An Y) electrical firm 1 0.10 0.15 (d) Find Var(X +atY50).ohms. The / / variconstant fixed However, is a random 2 are in use, respectively. jointI probability density bulb, which, according is given by 3 0.20 0.30 sewhere, able with µI =Y 15 and σI2 = 0.03 amperes2 . function for (X, ) isamperes given by 5 0.10 0.15 package, has a mean lif Give numerical approximations to the mean and vari4.71ance The length of time Y , in minutes, required to deviation of 50 hours. $ of the power 2 3 P2. 138 Chapter 4 M + yto ),tear 0 ≤gas x, has y ≤the 1, density the bulbs fail generate a human2 (x reflex to last eve f (x, y) = < y < 1, function 4.69 Consider0, Review Exerciseelsewhere. 3.77 on page 108. The distribution is symmetr 2and Y represent the number of verandom 1 X−y/4 sewhere. Find E(X)variables and !E(X ) and using these values, 4.65 Let X represent the n e ,separate 0then, ≤ ystreet < ∞,corners 4 at hicles fthat 2 twoor 2 2 (a) (y) arrive =whether red4.76 die is Seventy tossed and Y jobs the Determine not X and Y areduring indepennew evaluate E[(2X + 1) ]. m 4.7 to evaluate E(2XY − X Y ) a) Determine whether or not2-minute X and0,Yperiod are independent. a certain inelsewhere. the day. The joint distri- a green die is tossed. Find Zability = XYdistribution . bile manufacturing plan shown in Table dent. bution b) It is of interest to knowissomething about the proportion of Z = X+Y, the sum of the 4.58 The total time, % measured of 100 hours, (a)for E(X Y );positions. To the+70 & %in units (a) What isE(X+Y). the mean time to&reflex? two proportions. Find Sol: 5/4 1 9 that a teenager runs her hair dryer over a period of one the applicants, the com 2f (x, y) = , c) Alsovariables find E(XY ). is aE(Y Sol: (b) 3/8 E(X − Y ); (x+y) (b)year Find ) and Var(Y ). variable are independent random continuous random X that has the 16 4 mechanical skill, manua = 5 and σY2 = 3, find the variance d) Find Var(X),density Var(Y ),function and Cov(X, Y). Sol: 73/960, 73/960, -1/64 (c) E(XY ). ability. The mean grad for x = 0, 1, 2, . . . and y = 0, 1,Sol: 2, . .29/240 .. able Z = −2X + 4Ye) −Find 3. Var(X + Y). ⎧ 4.72 A manufacturing company has developed a ma- 60, and the scores have (a) Give E(X), E(Y Var(Y ). 4.66 Let X represent the n x, ), Var(X), 0 < xand 1, chine for cleaning ⎨ carpet that is < fuel-efficient becausegreen rcise 4.62 if X and Y are not indea person who and scores 84 is tossed Y the (b) Consider Z = X + Y , the sum of the two. Find EX 19: A manufacturing company has developed a machine for cleaning carpet that die is fuelf (x) = 2 − x, 1 ≤ x < 2, = 1. it delivers carpet cleaner so rapidly. Of interest is a jobs? [Hint: Use Cheby ⎩ a red Y, diethe is tossed. Find th and Var(Z). efficient because itrandom deliversE(Z) carpet cleaner so rapidly. Ofininterest a random variable elsewhere. variable Y , 0, the amount gallonsis per minute the distribution is symm variable amount in gallons per minute It delivered. It is known the density function is given by t X and Y are independent random delivered. is known that the that density function is107. given (a) 2X − Y ; Consider Review Exercise 3.64 ofonthe page Use4.70 Theorem 4.6 to evaluate the mean random bability densities and byvariable 2 A− random variab There are two service lines. The random variables X Y = 60X + 39X, where Y is equal to the (b)4.77 X + 3Y $8 25. ! and Y ofare the proportions of time that line 1 and line variance σ = 4. Using , x > 2, number kilowatt hours x3 1, expended 7The ≤ yjoint ≤ 8,annually. 2 are in use, respectively. probability density 0, elsewhere, f (y) = (a) If P (|X − 10|density ≥ 3); fun 4.67 the joint function for (X, Yvariable ) is 0,given elsewhere. 4.59 If a random Xbyis defined such that by (b) P (|X − 10| < 3); $3 2 2 2 2y ≤ 1, (x + y ), 0 ≤ x, (a) Sketch the density function. $ E[(X (c) P (5 $ <2X 15); 0 < f (x,−y)1)=] =2 10 and E[(X − 2) ] = 6, (x < + 2y), 2y, 0 < y <Give 1, E(Y), E(Y2), and Var(Y). Sol: 7.5, 169/3, 1/12 2 0, elsewhere. 7 f(d) (x, y) = value of the con (b) Give E(Y 2), E(Y ), and Var(Y ). the 0, els 0, elsewhere. find µ and σ . whether random or not Xvariables and Y are indepenEX 20: Suppose that X(a) andDetermine Y are independent having the joint probability dent. 4.73 For the situation inY are Exercise 4.72, random computefind the expected value of g( 4.60 Suppose that X and independent Y P (|X − E(e ) using Theorem 4.1,probability that is, bydistribution using variables having the joint " 8 4.68 The power P in watts x Y y electric circuit with resistan 4 =y) e 2f (y) dy. E(e f)(x, 2 value of Z = XY . distribution by P = I R, where I is curr 1 7 0.10 0.15 constant at 50 ohms. H 4.78 fixed Compute P (µ − y 3 0.20 0.30 Then compute E(eY ) 5not by using f (y), but rather by able with µ = 15 amperes I has the density function 0.10 0.15 using the second-order adjustment to the first-orderGive numerical approximati Find ance of the power P . approximation of E(eY ). Comment. Find (a) E(2X − 3Y ); ! 6x(1 4.69 Consider Review Exer− (b) E(XY ). 4.74 Consider again the situation of Exercise 4.72. It f (x) = a) E(2X − 3Y ) Sol: -2.60 0, Y r is required to find Var(eY ). Use Theorems 4.2 and 4.3random variables X and b) E(XY ) Sol: 9.60 hicles that arrive at two sepa 4.61 UseZTheorem 4.7 to use evaluate E(2XY 2 − of X 2ExerY ) a certain 2-minute period in the conditions and define = eY . Thus, for4.73 the to joint probability distribution shown in Table bution is cise find and compare with the r 3.1 on page 96. % theorem. Var(Z) = E(Z 2 ) − [E(Z)]2 . 1 f (x, y) = (x+y) 4.62 If X and Y are independent random variables 4 2 with variances σX = 5 and σY2 = 3, find the variance Review Exercises for x = 0, 1, 2, . . . and y = 0 of the random variable Z = −2X + 4Y − 3. (a) Give E(X), E(Y ), Var(X 4.63 Repeat Exercise 4.62 if X and Y are not inde(b) Consider Z = X + Y , t pendent andChebyshev’s σXY = 1. 4.79 Prove theorem. 4.81 Referring to the E(Z) and Var(Z). probability density fun Review Exercises 4.79 Prove Chebyshev’s theorem. 4.81 Referring to the r probability EX 21: Find the covariance of random variables X and Y having the joint probability density density funct 4.80 Find the covariance of random variables X and on page 105, find the ave function in the tank at the end of Y having the joint probability density function ! x + y, 0 < x < 1, 0 < y < 1, 4.82 Assume the length f (x, y) = 0, elsewhere. lar type of telephone conv Sol: -1/144 EX 22: As a student drives to school, he encounters a traffic signal. This traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school each weekday between 8:00 and 8:30 a.m. Let X1 be the number of times he encounters a green light, X2 be the number of times he encounters a yellow light, and X3 be the number of times he encounters a red light. Find the joint distribution of X1, X2, and X3. Sol: ? EX 23: Suppose that for a very large shipment of integrated-circuit chips, the probability of failure for any one chip is 0.10. Assuming that the assumptions underlying the binomial distributions are met, find the probability that at most 3 chips fail in a random sample of 20. Sol: 0.8670 EX 24: If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights a) exactly 18 will have a useful life of at least 800 hours; b) at least 15 will have a useful life of at least 800 hours; c) at least 2 will not have a useful life of at least 800 hours. Sol: 0.2852 Sol: 0.9887 Sol: 0.6083 EX 25: A manufacturer knows that on average 20% of the electric toasters produced require repairs within 1 year after they are sold. When 20 toasters are randomly selected, find appropriate numbers x and y such that a) the probability that at least x of them will require repairs is less than 0.5; Sol: 4 b) the probability that at least y of them will not require repairs is greater than 0.8. Sol: 14