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elementary probability for applications

Understanding Probability 2nd Edition Henk Tijms - Solutions

1. Problem 11.15 The amounts of rainfall in Amsterdam during each of the months January, February, ..., December are independent random variables with expected values of 62.1, 43.4, 58.9, 41.0, 48.3, 67.5, 65.8, 61.4, 82.1, 85.1, 89.0, and 74.9 mm and with standard deviations of 33.9, 27.8, 31.1,
1. Problem 11.14 Verify that var(aX +b) = a2var(X) and cov(aX,bY) = abcov(X,Y)var(Y) − ρ(X,Y)X/√for any constants a,b. Next, evaluate the variance of the random variable Z =Y/√var(X) to prove that −1 ≤ ρ(X,Y) ≤ 1. Also, for anyconstantsa,b,c,andd,verifythatcov(aX + bY,cV +
1.Problem11.13Thecontinuousrandomvariables X andY havethejointdensity f (x, y) = 4y2 for 0 < x < y < 1and f(x,y) = 0otherwise.Whatisthecor relation coefficient of X andY?Canyouintuitivelyexplainwhythiscorrelation coefficient is positive?
1. Problem 11.12 The independent random variables Z and Y have a standard normal distribution and a chi-square distribution with ν degrees of freedom.Use the transformation V = Y and W = Z/√variable W = Z/√Y/ν to prove that the random Y/ν has a Student-t density with ν degrees of
1. Problem 11.11 Let (X,Y) be a point chosen at random inside the unit circle.Define V and W by V = X√−2ln(Q)/Q and W = Y√−2ln(Q)/Q, where Q =X2+Y2.Verifythat the random variables V and W are independent and N(0,1) distributed. This method for generating normal variates is known as
1.Problem11.10 Apoint(V,W)ischoseninside the unit circle as follows. First, a number R is chosen at random between 0 and 1. Next, a point is chosen at randomonthecircumferenceofthecirclewithradius R.Usethetransformation formula to find the joint density function of this point (V,W). What is the
1. Problem 11.9 Thenonnegative random variables X and Y are independent and uniformly distributed on (c,d). What is the probability density of Z = X + Y?Whatistheprobabilitydensityfunctionof V = X2 + Y2?Usethelatterdensity to calculate the expectedvalueofthedistanceofapointchosenatrandominside the
1.Problem 11.8 The continuous random variables X and Y are nonnegative and independent. Verify that the density function of Z = X + Y is given by the convolution formula zfZ(z) =0 fX(z − y) fY(y)dy for z ≥ 0.
1.Problem 11.7 A point (X,Y) is chosen at random in the equilateral triangle having (0,0),(1,0), and (1 2 , 1 2 √ 3) as corner points. Determine the marginal densities of X and Y. Before determining the function fX(x), can you explain why fX(x) must be largest at x = 1 2 ?
1. Problem 11.6 Independently of each other, two numbers X and Y are chosen at random in the interval (0,1). Let Z = X/Y be the ratio of these two random numbers.(a) Use the joint density of X and Y to verify that P(Z ≤ z) equals 1 2z for 0
1. Problem 11.5 Independently of each other, two points are chosen at random in the interval (0,1). What is the joint probability density of the smallest and the largest of these two random numbers? What is the probability density of the length of the middle interval of the three intervals that
1. Problem 11.4 Let X and Y be two random variables with a joint probability density f (x, y) =1(x+y)3 0for x, y > c otherwise, for an appropriate constantc. Verify that c = 1 4and calculate the probability P(X >a,Y >b)fora,b > c.
1.Problem 11.3 Apoint (X,Y) is picked at random inside the triangle consisting of the points (x, y) in the plane with x, y ≥ 0 and x + y ≤ 1. What is the joint probability density of the point (X,Y)? Determine the probability density of each of the random variables X + Y and max(X,Y).
1. Problem11.2Let X denotethenumberofheartsandY thenumberofdiamonds in a bridge hand. What is the joint probability mass function of X and Y?
1.Problem11.1Yourollapairofdice.Whatisthejointprobabilitymassfunction of the low and high points rolled?
1.Problem10.27LetXbeacontinuousrandomvariablewithprobabilityden sityfunction f(x).Supposethat theprobabilitydistributionfunctionF(x)=P(X≤x)isstrictlyincreasingontherangeofX.DefinethefunctionI(u)as theinversefunctionofF(x).Verifythat(a) P(I(U)≤x)=P(X≤x)forallx,wherethecontinuousrandom
1.Problem10.26VerifythatarandomobservationfromtheWeibulldistribution withshapeparameterαandscaleparameterλcanbesimulatedby taking X=1λ[−ln(1−U)]1/α,whereUisarandomnumberfromtheinterval(0,1).Inparticular,X=−1λln(1−U)isarandomobservationfromtheexponential distributionwithparameterλ.
1.Problem10.25Apopulationofbacteriahas theinitial sizes0. Ineachgen eration, independentlyofeachother, it isequallylikelythat thepopulation increasesby25%ordecreasesby20%.What istheapproximateprobability densityofthesizeofthepopulationafterngenerationswithnlarge?
1. Problem 10.24 In order to prove that the normal probability density func tion integrates to 1 over the interval (−∞,∞), evaluate the integral I =∞−∞e−1 2 x2 dxforthestandardnormaldensity.Bychangingtopolarcoordinates inthedoubleintegral I2 = ∞∞−∞2−∞e−1(x2+y2)
1.Problem 10.23 Suppose that X1,...,Xn are independent random variables that are uniformly distributed on (0,1). What is the probability that the rounded sum X1 +···+Xn equals the sum of the rounded Xi when all rounding is to the nearest integer? Use the central limit theorem to verify that this
1.Problem 10.22 You perform an experiment that consists of ten independent Bernoulli trials. Before the experiment is done, your prior density of the suc cess probability of the Bernoulli trials is a beta density with parameters α and β. Argue that the beta density with parameters α∗ = α + 7
1.Problem10.21Usepropertiesofthegammafunctiontoderive E(X)and E(X2) for a gamma-distributed random variable X.
1. Problem 10.20 Let the random variables V and W be defined by V = √U and W =U2whenU isanumberchosenatrandombetween0and1.Whatarethe expected values and the standard deviations of V and W?
1.Problem 10.19 A stick of unit length is broken at random into two pieces. Let the random variable X represent the length of the shorter piece. What is the median of the random variable (1 − X)/X?
1. Problem 10.18 Suppose that the continuous random variable X has the prob ability density function f (x) = (α/β)(β/x)α+1 for x >βand f(x) = 0 for x ≤β for given values of the parameters α>0 and β>0. This density is called the Pareto density, which provides a useful probability model for
1. Problem 10.17 In an inventory system, a replenishment order is placed when the stock on hand of a certain product drops to the level s, where the reorder point s is a given positive number. The total demand for the product during the lead time of the replenishment order has the probability
1. Problem10.16Consider Problem 10.6 again. Calculate the expected value and standard deviation of the height above the ground when the ferris wheel stops.
1. Problem10.15LetXbeacontinuousrandomvariablewithprobabilitydensity f(x)andfiniteexpectedvalueE(X).10.1 Concept of probability density 295(a) What constant c minimizes E[(X − c)2] and what is the minimal value of E[(X −c)2]?(b) Prove that E(|X − c|) is minimal if c is chosen equal to the
1. Problem10.14Thelifetime(inmonths)ofabatteryisarandomvariableX satisfyingP(X≤x)=0forx
1.Problem10.13ApointQischosenatrandominsideaspherewithradiusr.Whataretheexpectedvalueandthestandarddeviationofthedistancefromthe centerofthespheretothepointQ?
1. Problem10.12Let X beanonnegativecontinuousrandomvariablewithdensity function f (x). Use an interchange of the order of integration to verify that E(X) = ∞0 P(X >u)du.
1. Problem10.11Apointischosenatrandominsideatrianglewithheighthand baseoflengthb.What istheexpectedvalueoftheperpendiculardistanceof thepointtothebase?10.1 Concept of probability density 293
1. Problem10.10Apoint ischosenat randominside theunit circle.Let the randomvariableVdenotetheabsolutevalueofthex-coordinateofthepoint.WhatistheexpectedvalueofV?
1. Problem10.9Apointischosenatrandominsidetheunitsquare{(x,y):0≤x,y≤1}.Whatistheexpectedvalueofthedistancefromthispointtothepoint(0,0)?
1. Problem10.8InanInternetauctionofacollector’sitemtenbidsaredone.The bidsareindependentofeachotherandareuniformlydistributedon(0,1).The personwiththelargestbidgetstheitemforthepriceofthesecondlargestbid(a so-calledVickreyauction).Arguethattheprobabilityofthesecondlargestbid exceedingthevaluex
1.Problem10.7The javelin throwerBigJohn throws the javelinmore than xmeterswithprobabilityP(x),where P(x)=1 for 0≤x
1. Problem 10.6 Suppose you decide to take a ride on the ferris wheel at an amusement park. The ferris wheel has a diameter of 30 meters. After several turns, the ferris wheel suddenly stops due to a power outage. What random variable determines your height above the ground when the ferris wheel
1.Problem 10.5 The numbers U1 and U2 are chosen at random from the interval(0, 1), independently of each other. Let the random variables V and W be defined by V = min(U1,U2) and W = max(U1,U2). What are the probability density functions of the random variables V and W?10.1 Concept of probability
1. Problem 10.4 A stick of unit length is broken at random into two pieces. Let the random variable X represent the length of the shorter piece. What is the probability density of X? Also, use the probability distribution function of X to give an alternative derivation of the probability density of
1.Problem 10.3 The number X is chosen at random between 0 and 1. Determine theprobabilitydensityfunctionofeachoftherandomvariablesV = X/(1 − X)and W = X(1− X).
1. Problem 10.2 A point Q is chosen at random inside the unit square. What is the density function of the sum of the coordinates of the point Q? What is the density function of the product of the coordinates of the point Q? Use geometry to find these densities.
1.Problem 10.1 Let X be a positive random variable with probability density function f (x).DefinetherandomvariableY byY = X2.Whatistheprobability density function of Y? Also, find the density function of the random variable W =V2 if V is a number chosen at random from the interval (−a,a) with a
1. Problem 9.34 In the Lotto 6/45 six different numbers are drawn at random from the numbers 1,2,...,45. What are the probability mass functions of the largest number drawn and the smallest number drawn?
1. Problem 9.33 An absent-minded professor has m matches in his right pocket and m matches in his left pocket. Each time he needs a match, he reaches for a match in his left pocket with probability p and in his right pocket with probability 1 − p. When the professor first discovers that one of
1. Problem 9.32 In European roulette the ball lands on one of the numbers 0, 1,...,36 in every spin of the wheel. A gambler offers at even odds the bet that the house number 0 will come up once in every 25 spins of the wheel.What is the gambler’s expected profit per dollar bet?
1. Problem 9.31 A psychologist claims that he can determine from a person’s handwriting whether the person is left-handed or not. You do not believe the psychologist and therefore present him with 50 handwriting samples, of which 25 were written by left-handed people and 25 were written by
1. Problem 9.30 For a final exam, your professor gives you a list of 15 items to study. He indicates that he will choose eight for the actual exam. You will be required to answer five of those. You decide to study 10 of the 15 items. What is the probability that you will pass for the exam?
1. Problem 9.29 There is a concert and 2,500 tickets are to be raffled off. You havesentin100applications.Thetotalnumberofapplicationsis125,000.What are your chances of getting a ticket? Can you explain why this probability is approximately equal to 1 − e−2?
1. Problem 9.28 Ten identical pairs of shoes are jumbled together in one large box. Without looking, someone picks four shoes out of the box. What is the probability that, among the four shoes chosen, there will be both a left and a right shoe?
1. Problem 9.27 In the famous problem of Chevalier de M´er´e, players bet first on the probability that a six will turn up at least one time in four rolls of a fair die;subsequently, players bet on the probability that a double six will turn up in 24 rolls of a pair of fair dice. In a generalized
1. Problem 9.26 A die is rolled until a six appears for the third time. What is the probability distribution of the number of rolls required?
1. Problem 9.25 In the World Series Baseball, the final two teams play a series consisting of a possible seven games until such time that one of the two teams has won four games. In one such final, two unevenly matched teams are pitted against each other and the probability that the weaker team
1. Problem 9.24 Onbridge night, the cards are dealt round seven times. Only two times do you receive an ace. From the beginning, you had your doubts as to whether the cards were being shuffled thoroughly. Are these doubts confirmed?
1. Problem 9.23 What is the fewest number of dice one can roll such that, when they are rolled simultaneously, there will be at least a 50% probability of rolling two or more sixes?282 Basic rules for discrete random variables
1.Problem 9.22 Daily Airlines flies every day from Amsterdam to London. The price for a ticket on this popular route is $75. The aircraft has a capacity of 150 passengers. Demand for tickets is greater than capacity, and tickets are sold out well in advance of flight departures. The airline company
1. Problem9.21Yourepeatedlydrawarandomintegerfromtheintegers1,...,10 until you have three different integers. What is the probability that you need r draws?
1.Problem 9.20 Modify the convolution formula in Rule 9.9 when the random variables X and Y are integer-valued but not necessarily nonnegative.
1. Problem 9.19 Let Xi denote the number of integers smaller than i that precede i in a random permutation of the integers 1,...,10. What are the expected value and the variance of the sum X2 +···+X10?
1. Problem9.18Adrunkardisstanding in the middle of a very large town square.Hebeginstowalk.Eachstepisaunitdistance in oneofthefourdirections East, West, North, and South. All four possible directions are equally probable. The direction for each step is chosen independently of the direction of the
1. Problem 9.17 Two fair dice are tossed. Let the random variable X denote the sumof the two numbers shown by the dice and let Y be the largest of these two numbers. Are the random variables X and Y independent? What are the values of E(XY) and E(X)E(Y)?
1.Problem 9.16 Let X andY betwoindependentrandomvariables. What are the expected value and the variance of X − Y?
1. Problem 9.15 The University of Gotham City renegotiates its maintenance contract with a particular copy machine distributor on a yearly basis. For the coming year, the distributor has come up with the following offer. For a prepaid cost of $50 per repair call, the university can opt for a fixed
1. Problem9.14 Atthe beginning of every month, a pharmacist orders an amount of a certain costly medicine that comes in strips of individually packed tablets.The wholesale price per strip is $100, and the retail price per strip is $400.The medicine has a limited shelf life. Strips not purchased by
1. Problem 9.13 Consider Example 9.3 again. What is the standard deviation of the number of trials required?
1.Problem9.12 Calculate the standard deviation of the random variables appear ing in the Examples 9.1 and 9.2.
1. Problem 9.11 What is the expected value of the number of combinations of two consecutive numbers in a lottto drawing of six different numbers from the numbers 1,2,...,45?
1. Problem 9.10 What is the expected value of the number of times that two adjacent letters are the same in a random permutation of the word Mississippi?
1. Problem 9.9 What is the expected number of distinct birthdays within a ran domly formed group of 100 persons?
1.Problem 9.8 Consider Example 7.13 again. Calculate the expected number of hotels that remain empty. Hint: define the random variable Xi as equal to 1 if the ith hotel remains empty and 0 otherwise.
1. Problem9.7Supposethat therandomvariableXisnonnegativeandinteger valued.VerifythatE(X)= ∞k=0P(X>k).
1. Problem 9.6 Mary and Peter play the following game. They toss a fair coin until heads appears for the first time or m tosses are done, whichever occurs first. Here m is fixed in advance. If heads appears at the kth toss, then Peter pays Mary 2k dollars when k is odd and otherwise Mary pays Peter
1. Problem 9.5 A stick is broken at random into two pieces. You bet on the ratio of the length of the longer piece to the length of the smaller piece. You receive $k if the ratio is between k and k + 1 for some k with 1 ≤ k ≤ m −1, while you receive $m if the ratio is larger than m. Here m is
1. Problem 9.4 In a lottery, one thousand tickets numbered as 000,001,...,999 are sold. Each contestant buys only one ticket. The prize winners of the lot tery are determined by drawing at random one number from the numbers 000,001,...,999. You are a prize winner when the number on your ticket is
1. Problem9.3Youspinagameboardspinnerwith1,000equalsectionsnumbered as 1,2,...,1,000. After your first spin, you have to decide whether to spin the spinner for a second time. Your payoff is the total score of your spins as long as this score does not exceed 1,000; otherwise, your payoff is zero.
1. Problem 9.2 Calculate the expected value of the greater of two numbers when two different numbers are picked at random from the numbers 1,...,n. What is the expected value of the absolute difference between the two numbers?
1.Problem 9.1 You are playing a game in which four fair dice are rolled. A $1 stake is required. The payoff is $100 if all four dice show the same number 9.2 Expected value 267 and $10 if two dice show the same even or odd number. What is an appropriate probability space for this experiment, and
1. Problem8.18Afriendlycoupletellsyouthattheydida100%reliablesonogram test and found out that they are going to have twin boys. They asked the doctor about the probability of identical twins rather than fraternal twins. The doctor could only give themtheinformation that the population proportion of
1. Problem 8.17 You have five coins colored red, blue, white, green, and yellow.Apart from the variation in color, the coins look identical. One of the coins 256 Conditional probability and Bayes is unfair and when tossed comes up heads with a probability of 3 4; the other four are fair coins. You
1.Problem 8.16 In a certain region, it rains on average once in every ten days during the summer. Rain is predicted on average for 85% of the days when rainfall actually occurs, while rain is predicted on average for 25% of the days when it does not rain. Assume that rain is predicted for tomorrow.
1. Problem 8.15 Consider the lost boarding pass puzzle that was stated in Sec tion 2.9.2. Assume now that N people are lining up to board an airplane with N seats. Verify that the probability of the last passenger getting his/her own seat equals 1 2, regardless of the value of N ≥ 2.
1.Problem 8.14 It is believed that a sought-after wreck will be in a certain sea area with probability p = 0.4. A search in that area will detect the wreck with probability d = 0.9 if it is there. What is the revised probability of the wreck being in the area when the area is searched and no wreck
1. Problem 8.13 A fair coin is tossed k times. Let ak denote the probability of havingnotwoheadsinarowinthesequenceoftosses.Usethelawofconditional probabilities to obtain a recurrence relation for ak. Calculate ak for k = 5, 10, 25, and 50.
1. Problem 8.12 A fair die is rolled repeatedly. Let pn be the probability that the sum of scores will ever be n. Use the law of conditional probabilities to find a recursion equation for pn. Verify numerically that pn tends to 1 3.5= 0.2857 as n gets large. Can you explain this result?
1. Problem 8.11 Consider the scratch-lottery problem from Section 4.2.3. Each weekonemillion scratch-lottery tickets are printed. Assume that in a particular week only one-half of the tickets printed are sold. What is the probability of at least one winner in that week? Hint: use results from
1. Problem8.10Let’sreturntothecasinogameRedDogfromProblem3.25.Usingthelawofconditionalprobabilities,calculatetheprobabilityoftheplayer 8.2 Bayes’ rule in odds form 251 winning. Hint: argue first that the probability of a spread of i points is given by 152![(12 − i) × 4 ×4×2].
1. Problem8.9Threefriendstravelbyplaneforthefirsttime.Theygotassigned theseatsA(window),B(middle),andC(aisle)inthesamerow.OntheseatsA andCpassengerscannotwronglyfastentheirseatbelts,butaninexperienced traveleronthemiddleseatBfasteningtheseatbeltfirsthasa50-50chance
1.Problem8.8Adieisrolledtoyieldanumberbetween1and6,andthenacoin istossedthatmanytimes.Whatistheprobabilitythatheadswillnotappear?
1.Problem 8.7 Two fair coins are tossed. Let A be the event that heads appears on the first coin and let B be the event that the coins display the same outcome. Are the events A and B independent?
1. Problem8.6Yourfavoriteteamparticipatesinaknock-outsystemthatconsists offourrounds.Ifyourteamhasreachedroundi,itwillsurvivethisroundwitha givenprobabilityofpi fori=1,...,4.Afterthecompetition,youareinformed thatyourteamisnotthefinalwinner.Thisistheonlyinformationyougetabout
1. Problem8.5Sevenindividualshavereservedticketsat theopera.Theseats theyhavebeenassignedareall inthesamerowofsevenseats.Therowof seats isaccessiblefromeitherend.Assumethat thesevenindividualsarrive andtake their seats inarandomorder.What is theprobabilityofall seven
1.Problem8.4YoutravelfromAmsterdamtoSidneywithchangeofairplanesin DubaiandSingapore.Youhaveonepieceofluggage.Ateachstopyourluggage istransferredfromoneairplanetoanother.AttheairportinAmsterdamthere isaprobabilityof5%thatyourluggageisnotplacedintherightplane.This probabilityis3%at theairport
1. Problem 8.3 Suppose a bridge player’s hand of 13 cards contains an ace. What is the probability that the player has only one ace? What is the answer to this question if you know that the player had the ace of hearts?
1. Problem 8.2 Someone has tossed a fair coin three times. You know that one of the tosses came up heads. What is the probability that at least one of the other two tosses came up heads as well?
1.Problem 8.1 Every evening, two weather stations issue a weather forecast for the next day. The weather forecasts of the two stations are independent of each other. On average, the weather forecast of station 1 is correct in 90% of the cases, irrespective of the weather type. This percentage is
1. Problem 7.23 What is the probability that a hand of 13 cards contains four of a kind?
1. Problem 7.22 Consider the card game Jeu de Treize from Problem 3.32. Use the inclusion-exclusion rule to verify that the probability of the dealer winning the first round is 0.6431.
1.Problem7.21Whatistheprobability that in a player’s hand of 13 cards at least one suit will be missing?
1. Problem7.20 Forthe upcomingdrawing of the Bingo Lottery, five extra prizes have been addedtothe pot. Each prize consists of an all-expenses paid vacation trip. Each prize winner may choose from among three possible destinations A, B, and C. The three destinations are equally popular. The prize
1. Problem 7.19 An integer is chosen at random from the integers 1,...,1000.What is the probability that the integer chosen is divisible by 3 or 5? What is the probability that the integer chosen is divisible by 3, 5 or 7?
1. Problem 7.18 In the casino game of Chuck-a-Luck, three dice are contained within an hourglass-shaped, rotating cage. You bet on one of the six possible numbers and the cage is rotated. You lose money only if your number does not come up on any of the three dice. Much to the pleasure of the

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