Understanding Probability 2nd Edition Henk Tijms - Solutions

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1.4. For a general increasing function F there is a similar decomposition F -Fe + Fd, where both Fe and Fd are IncreasIng, Fe IS contInUOUS, and F d is "purely jumping". [HINT: Let a be a point of continuity, put Ed (a)F(a), add jumps in (a, (0) and subtract jumps in (-00,a) to define Fd.
1.3. A plausible verbal definition of a discrete dJ. may be given thus: "It is a d.f. that has jumps and is constant between jumps." [Such a function is sometimes called a "step function", though the meaning of this term does not seem to be well established.] What is wrong with this? But suppose
1.* 2. Let F be a d.f. with points of jump {a j}. Prove that the sum X-E
1.1. Let F be a dJ. Then for each x, lim[F(x + E) -F(x -E)] = 0 EtO unless x is a point of jump of F, in which case the limit is equal to the size of the jump.
1. Problem14.15Themoment-generatingfunction of two jointly distributed ran dom variables X and Y is defined by MX,Y(v,w) = E(evX+wY), provided that this integral is finite for all (v,w) in a neighborhood of (0,0). A basic result is that MX,Y(v,w) uniquely determines the joint distribution of X and
1. Problem 14.14 The random variables X and Y are independent and identically distributed. Use the moment-generating function to prove that X and Y are normallydistributedif X + Y hasanormaldistribution.Remark:theassumption that X and Y are identically distributed can be dropped, but this requires
1.Problem 14.13 Let X1,...,Xn be independent random variables each having a gamma density with the same scale parameter β. Denote by αi the shape parameter of the gamma density of the Xi. Verify that X1 +···+Xn has a gamma density with shape parameter α1 +···+αn and scale parameter β.
1..Problem 14.12 Determine the moment-generating function of the random variable X having as density function the so-called Laplace density function f (x) = 1 2ae−a|x| for −∞ < x < ∞, where a is a positive constant. Use the moment-generating function of X to find E(X) and var(X).
1. Problem 15.37 Let c(i) be a given function on a finite set I. For any i ∈ I,a local neighborhood N(i) of other points from I is given such that k ∈ N(j)if j ∈ N(k). For ease, it is assumed that each set N(i) contains the same number of points. The following Markov chain is defined on I. If
1. Problem 15.36 Let {rj, j ∈ I} be a given probability distribution, where I is a finite set with N elements. Define the Markov chain {Xn} on the state space I by the one-step transition probabilities pij = 1 N−1 min(rj ri, 1) for j= i and 414 Markov chains pii = 1− j:j=i pij.† Prove that
1. Problem 15.35 Consider Problem 15.1 again. Let the Markov chain {Xn}describe the number of type-1 particles in compartment A. Prove that the equi librium probabilities satisfy the recurrence relation pk,k−1πk = pk−1,kπk−1 Use this result to verify that πj = r jr for k = 1,2,...,r.r−j/
1. Problem15.34 Atransport firm has effected an insurance contract for a fleet of vehicles. The premium payment is due at the beginning of each year. There are four possible premium classes with a premium payment of Pi in class i, where Pi+1 < Pi fori =
1. Problem 15.33 Consider Problem 15.6 again. Suppose that a cost of $750 is incurred each time the device fails and that each circuit board replaced costs 15.4 Long-run analysis of Markov chains 413$100. What is the long-run proportion of weeks the device operates properly?What is the long-run
1. Problem 15.32 Consider Problem 15.4 again. The local entrepeneur Jerry Woodside has a restaurant on the island. On every sunny day, his turnover (in dollars) has an N(μ1,σ2 1) distribution with μ1 = 1,000 and σ1 = 200, while onrainy dayshisturnover is N(μ2,σ2 2) distributed with μ2 =
1. Problem 15.31 Consider Example 15.4 again. What is the long-run average stock on hand at the end of the week? What is the long-run average ordering frequency and what is the long-run amount of demand lost per week?
1. Problem15.30ConsiderProblem2.44fromChapter2withParrondo’sparadox again. For each of the two strategies described in this problem, use a Markov chain to calculate the long-run win probability. Hint: use a Markov chain with three states and a Markov chain with 12 states.
1. Problem 15.29 Let {Xn} be a Markov chain with no two disjoint closed sets and state space I ={1,2,...,N}. Suppose that the Markov chain is doubly stochastic; that is, for each of the columns of the matrix of one-step transition probabilities the column elements sum to one. Verify that the Markov
1. Problem 15.28 Consider Example 15.7 again. It is now assumed that Boris Karparoff and Deep Blue play infinitely often against each other. What is the long-run proportion of games won by Boris? How often will Boris win a game after having won the previous game?
1. Problem 15.27 Consider Example 5.3 again. What is the long-run proportion of time the professor has his license with him? Also, answer this question for Problem 15.2.
1. Problem 15.26 Consider Problem 15.4 again. Calculate the equilibrium prob abilities of the Markov chain describing the weather. What is the long-run proportion of days it will be sunny? What is the probability that it will be rainy on a given Sunday many days from now?
1.Problem 15.25 The much feared professor Frank N. Stone gives varying ver sions of an oral examination in assembly line fashion, with students taking the exam one after the other. Each version of the exam may be categorized as difficult, normal or easy. After a difficult exam, the next exam will
1. Problem 15.24 Consider Problem 15.4 again. Use an absorbing Markov chain to calculate the probability of having no rain on two consecutive days during the next seven days given that it was sunny during the last two days. What is the value of this probability if the last two days were rainy?
1. Problem 15.23 Consider Problem 15.8 again. A certain car is now at location 4. As soon as this car returns to location 1, it will be overhauled. What is the probability that the car will be rented out more than five times before it returns to location 1? What is this probability if the car is
1. Problem 15.22 Consider Problem 2.40 from Chapter 2 again. Use a Markov chain to find the probability of the first passenger in line changing seats r or more times before getting to his assigned seat for r = 1,2,...,10. What is the expected number of times the passenger will change seats? Also,
1. Problem 15.21 You start out with 25 coins in small change in your pocket.Each time a beggar asks you for money you give him a random number of the coins left in your pocket. Use an absorbing Markov chain to calculate the probability mass function of the number of beggars who are favored by
1. Problem 15.20 Joe Dalton desperately wants to raise his current bankroll of$800 to $1,000 in order to pay his debts before midnight. He enters a casino and decides to play for high stakes at European roulette. He bets on red each time. The stake is $200 if his bankroll is $200 or $800 and is
1. Problem 15.19 In European roulette the wheel is divided in 37 sections, num bered as 1,2,...,36 and 0. Of the sections numbered from 1 to 36, 18 are red and18areblack.Thesectionmarked0iswinningforthehouse.Useanabsorb ing Markov chain to calculate the probability that in the next 1,000,000 spins
1. Problem 15.18 A fair coin is tossed until the last three tosses either show the combination TTH or the combination THH. Here H stands for heads and T stands for tails. What is the probability that the combination TTH appears before the combination THH? Can you explain why this probability is
1. Problem 15.17 The Bubble Company offers a picture of one of 25 popstars in a pack of chewing gum. John and Peter each buy one pack every week. They pool the pictures of the popstars. Assuming equal chances of getting any of the 25 pictures with one purchase, denote by the random variable N the
1. Problem 15.16 Calculate the probability of a run of five heads or five tails occurring in 20 tosses of a fair coin. What is the probability of a run of five heads occurring in 20 tosses of a fair coin?
1. Problem 15.15 In each drawing of the Lotto 6/45 six different numbers are drawn from the numbers 1,2,...,45. Calculate for r = 15, 25, 35, and 50 the probability that more than r drawings are needed until each of the numbers 1, 2,...,45 has been drawn.
1.Problem 15.14 A theater buff has attended 150 performances at a theater with 49 seats. At the start of each performance, the theater buff has been randomly directed to one of the 49 seats. Calculate the probability that this person has occupied every seat in the theater at least one time.
1. Problem 15.13 Let {Xn} be any Markov chain. For any i and j, define the random variable Vij(n) as the number of visits to state j over the time points t =1,2,...,n if the starting state is i. Verify the result nσ2[Vij(n)] =t=1 np(t)ij 1− p(t)ij +2 t=1 nu=t+1 p(t)ij p(u−t)jj Hint: use the
1. Problem 15.12 Trees in a forest are assumed to fall into four age groups: baby trees (0-10 years of age), young trees (11-20 years of age), middle-aged trees(21-30 years of age), and old trees (older than 30 years of age). The length of one time period is 10 years. In each time period a certain
1. Problem 15.11 A faulty digital video conferencing system has a clustering error pattern. If a bit is received correctly, the probability of receiving the next bit correctly is 0.999. This probability is only 0.1 if the last bit was received incorrectly. Suppose that the first transmitted bit is
1. Problem 15.10 A communication system is either in the on-state (state 1) or the off-state (state 0). Every millisecond the state of the system may change.An off-state is changed into an on-state with probability α and an on-state is changedintoanoff-statewithprobabilityβ,where0
1. Problem 15.9 Consider Problem 15.4 again. What is the probability of having sunny weather five days from now if it rained today and yesterday? What is the proportion of time it will be sunny over a very long period? What is the expected number of days it will be sunny in the next 14 days given
1.Problem15.8Acar rentalagencyrentscarsat four locations.Arentedcar canbereturnedtoanyofthefourlocations.Acarrentedatlocation1willbe returnedtolocation1withprobability0.8, tolocation2withprobability0.1, tolocation3withprobability0,andtolocation4withprobability0.1.These 15.2 Transient analysis of
1. Problem15.7Acommunicationchanneltransmits messages one at a time, and transmission of a message can only start at the beginning of a time slot. The transmission time of any message is one time slot. However, each transmission can fail with a given probability f = 0.05. A failed transmission is
1. Problem 15.6 A control device contains two parallel circuit boards. Both cir cuit boards are switched on. The device operates properly as long as at least one of the circuit boards functions. Each circuit board is subject to random failure. The failure rate increases with the age of the circuit
1. Problem 15.5 To improve the reliability of a production system, two identical production machines are connected parallel to one another. For the production process, only one of the two machines is needed. The other machine (if avail able) takes over when the machine currently in use needs
1. Problem 15.4 Every day, it is either sunny or rainy on Rainbow Island. The weather for the next day depends only on today’s weather and yesterday’s weather. If the last two days were sunny, it will be sunny on the next day with probability 0.9. This probability is 0.45 if the last two days
1. Problem 15.3 Let {Xn, n = 0,1,...} be a Markov chain. Define the random variables Yn and Un by Yn = X2n and Un = |Xn|. Do you think the processes{Yn} and {Un} are always Markov chains?
1. Problem 15.2 Consider the following modification of Example 15.3. In the case that the driver’s license of the professor is at his point of departure, he takes it with him with probability 0.75 when departing from home and with probability 0.5 when departing from the office. Define a Markov
1.Problem15.1Twocompartments A and B eachcontainr particles.Ofthese2r particles, r are of type 1 andr are of type 2. With the passing of every time unit, one particle is selected at random from each of the compartments, and each of these two particles is transferred from its compartment to the
1. Problem14.18Let therandomvariableXbethenumberofsuccesses inn independentBernoullitrialswithsuccessprobabilityp.Chooseanyδ>0such that(1+δ)p
1.Problem14.17ProvethatP(X≤c)≤mint
1. Problem 14.16 Consider Problem 14.5 again, but assume now that the indi vidual claim sizes have a gamma density with shape parameter α and scale parameter λ. What is the moment-generating function of the total claim size?
1. Problem14.15Themoment-generatingfunction of two jointly distributed ran dom variables X and Y is defined by MX,Y(v,w) = E(evX+wY), provided that this integral is finite for all (v,w) in a neighborhood of (0,0). A basic result is that MX,Y(v,w) uniquely determines the joint distribution of X and
1. Problem 14.14 The random variables X and Y are independent and identically distributed. Use the moment-generating function to prove that X and Y are normallydistributedif X + Y hasanormaldistribution.Remark:theassumption that X and Y are identically distributed can be dropped, but this requires
1. Problem 14.13 Let X1,...,Xn be independent random variables each having a gamma density with the same scale parameter β. Denote by αi the shape parameter of the gamma density of the Xi. Verify that X1 +···+Xn has a gamma density with shape parameter α1 +···+αn and scale parameter β.
1.Problem 14.12 Determine the moment-generating function of the random variable X having as density function the so-called Laplace density function f (x) = 1 2ae−a|x| for −∞ < x < ∞, where a is a positive constant. Use the moment-generating function of X to find E(X) and var(X).
1. Problem14.11Apopulationofbacteriabeginswithasingleindividual.Ineach generation, each individual dies with probability 1 3or splits in two with proba bility 2 3. What is the probability that the population will die out by generation 3 and what is the probability that the population will die out
1.Problem14.10Everyadultmaleinacertainsocietyismarried. Twentypercent of the married couples have no children. The other 80% have two or three children with respective probabilities 1 3and 2 3, each child being equally likely to be a boy or a girl. What is the probability that the male line of a
1. Problem 14.9 Let X be the number of tosses of a fair coin until the number of heads first exceeds the number of tails. Use the method of first-step analysis to obtain the generating function of X. What is the expected value of X?
1. Problem 14.8 Independently of each other, you generate integers at random from 0,1,...,9 until a zero is obtained. Use the method of first-step analysis to obtain the generating function of the sum of the generated integers.
1.Problem14.7Youparticipate in a game that consists of a series of independent plays. Any play results in a win with probability p, in a loss with probabil ity q and in a draw with probability r, where p +q +r = 1. One point is added to your score each time a play is won; otherwise, your score
1. Problem 14.6 The number of customers asking for a new product has a Poisson distribution with expected value μ. The demands of the customers are independent random variables each having the probability mass func tion ak =−αk[k ln(1 −α)]−1, k = 1,2,... for a given 0
1. Problem 14.5 The number of claims an insurance company will receive is a random variable N having a Poisson distribution with expected value μ.The claim sizes are independent random variables with a common probabil ity mass function ak for k = 1,2.... Let the total claim size S be defined by S
1. Problem 14.4 The nonnegative, integer-valued random variables X and Y are independent and identically distributed. Verify that X and Y are Poisson dis tributed when the sum X + Y is Poisson distributed.
1. Problem14.3Supposethatyoudrawanumberatrandomfromtheunitinterval r times. A draw is called a “record draw” when the resulting number is larger than the previously drawn numbers. Determine the generating function of the numberofrecorddraws.Whataretheexpectedvalueandvarianceofthenumber of
1. Problem 14.2 Use the results of Problem 14.1 to obtain the expected value and the variance of both the binomial distribution and the negative binomial distribution.
1.Problem 14.1 Suppose that the random variable X has a binomial distribution with parameters n and p. Use the fact that X can be represented as the sum of n independent Bernoulli variables to derive the generating function of X.Ina similar way, derive the generating function of the random variable
1.Problem13.16Fixanumbera with0 < a < 1.Youdrawrepeatedlyarandom number from an interval until you obtain a random number belowa. The first random number is chosen from the interval (0,1) and each subsequent random number is chosen from the interval between zero and the previously chosen random
1.Problem 13.15 You spin a game board spinner in a round box whose circum ference is marked with a scale from 0 to 1. When the spinner comes to rest, it points to a random number between 0 and 1. After your first spin, you have to decide whether to spin the spinner for a second time. Your payoff is
1. Problem 13.14 In the game of Pig each player’s turn consists of repeatedly rolling a die. If the player rolls a 1, the player’s turn ends and nothing is added to the player’s score. If the player rolls a number other than 1, the player’s turn continues and the player has the choice
1. Problem 13.13 A fair coin is tossed no more than n times, where n is fixed in advance. You stop the coin-toss experiment as soon as the proportion of heads exceeds 1 2or as soon as n tosses are done, whichever occurs first. Use the law of conditional expectations to calculate, for n = 5, 10, 25,
1. Problem 13.12 Let’s return to Problem 13.6. Use the law of conditional expec tations to verify that the expected value of your waiting time until the next bus arrival is equal to 15.e
1. Problem 13.11 Consider the casino game Red Dog from Problem 3.25 again.Suppose that the initial stake of the player is $10. What are the expected values of the total amount staked and the payout in any given play? Use the law of conditional expectations to find these expected values.366
1. Problem 13.10 Nobel airlines has a direct flight from Amsterdam to Palermo.This particular flight uses an aircraft with N = 150seats.Thenumberofpeople who seek to reserve seats for a given flight has a Poisson distribution with expectedvalueλ = 170.Theairlinemanagementhasdecidedtobookupto Q
1. Problem 13.9 A farming operation is located in a remote area that is more or less unreachable in the winter. As early as September, the farmer must order fuel oil for the coming winter. The amount of fuel oil he needs each winter is random, and depends on the severity of the winter weather to
1.Problem 13.8 The percentage of zinc content and iron content in ore from a certain location has the joint density f (x, y) = 1 350(5x + y −30) for 2 < x
1. Problem 13.7 Suppose that U and Y are independent random variables, where U isuniformly distributed on (0,1) and Y is a continuous random variable with probability density g(y). Let f (y) be another probability density function such that, for some constant c (≥ 1), f (y) ≤ cg(y) for all y.
1. Problem13.6Youleaveworkatrandomtimesbetween5.45p.m.and6.00p.m.to take the bus home. Bus numbers 1 and 3 bring you home. You take the first bus that arrives. Bus number 1 arrives exactly every 15 minutes starting from the hour, whereas bus number 3 arrives according to a Poisson process with the
1. Problem 13.5 Use twice the law of conditional probabilities to calculate the probability that the quadratic equation Ax2 + Bx + C = 0 will have two real roots when A, B, andC areindependent samples from the uniform distribution on (0,1).
1. Problem 13.4 The random variables X1 and X2 are N(μ1,σ2 1) and N(μ2,σ2 2)distributed. Let the random variable V be distributed as X1 with given proba bility p and as X2 with probability 1 − p. What is the probability density of V?Is this probability density the same as the probability
1. Problem 13.3 Anopaque bowl contains B balls, where B is given. Each ball is red orwhite.Thenumberofredballsinthebowlisunknown,buthasabinomial distribution with parameters B and p. You randomly select r balls out of the urn without replacing any. Use the law of conditional probabilities to obtain
1.Problem 13.2 Consider the three-players variant of Example 13.6. Calculate the optimal strategy for the first player A, assuming that the other two players B and C play optimally. What is the probability distribution function of the final 360 Conditional distributions score of player A under his
1.Problem 13.1 The length of time required to unload a ship has an N(μ,σ2)distribution. The crane to unload the ship has just been overhauled and the time it will operate until the next breakdown has an exponential distribution with an expected value of 1/λ. What is the probability of no
1. Problem 12.16 Astudy of D. Kadell and D. Ylvisaker entitled “Lotto play: the good, the fair and the truly awful,” Chance Magazine 4 (1991): 22–25 analyzes the behavior of players in the lotto. They took 111,221,666 tickets that were manually filled in for a specific draw of the California
1. Problem12.15IntheDutchLottosixso-calledmainnumbersandoneso-called bonus number are drawn from the numbers 1,...,45 and in addition one color is drawn from six differing colors. For each ticket you are asked to mark six distinct numbers and one color. You win the jackpot (first prize) by matching
1. Problem 12.14 Use a chi-square test to investigate the quality of the random number generator on your computer. Generate 10,000 random numbers and count the frequencies of the numbers falling in each of the intervals i−1 10, i 10 for i = 1,...,10.
1.Problem 12.13 In a classical experiment, Gregor Mendel observed the shape and color of peas that resulted from certain crossbreedings. A sample of 556 peas was studied with the result that 315 produced round yellow, 108 produced round green, 101 produced wrinkled yellow, and 32 produced wrinkled
1.Problem 12.12 For the Lotto 6/45, simulate from the multivariate normal distribution in order to find approximately the probability P max 1≤j≤s Snj − min 1≤j≤s Snj > 1.5√ n forn=100,300,and500. Use computer simulation to find the exact value of this probability.
1. Problem 12.11 The random vector (X1, X2, X3) has a trivariate normal distri bution. What is the joint distribution of X1 and X2?
1.Problem 12.10 The annual rates of return on the three stocks A, B, and C have a trivariate normal distribution. The rate of return on stock A has expected value 7.5% and standard deviation 7%, the rate of return on stock B has expected value 10% and standard deviation 12%, and the rate of return
1. Problem12.9Twoparticlescarryoutadrunkard’swalkonthetwo-dimensional plane,independentlyofeachother.Bothparticlesstartattheorigin(0,0).One particleperformsn stepsand theothermsteps.Canyougivean intuitive explanationwhytheexpecteddistancebetweenthefinalpositionsofthetwo particlesisequalto1
1.Problem12.8Therectangulardistancefromtheorigintothepositionofthe particleafternstepsisdefinedbyRn=|Sn1|+|Sn2|.VerifythatE(Rn)≈ 4nπandRnhastheapproximatedensity 4 √2πn e−1 4r2/n[ ( r √n)− (−r √n)]fornlarge, where (x)isthestandardnormaldistributionfunction.
1. Problem 12.7 Let (X,Y) have a bivariate normal distribution with σ2(X) =σ2(Y). Prove that the random variables X + Y and X − Y are independent and normally distributed
1. Problem 12.6 The random variables Z1 and Z2 are independent and N(0,1)distributed. Define the random variables X1 and X2 by X1 = μ1 + σ1Z1 and X2 =μ2+σ2ρZ1 +σ2 1 −ρ2Z2, where μ1, μ2, σ1, σ2 and ρ are constants withσ1 > 0,σ2 > 0and−1
1. Problem 12.5 Let (X,Y) have a bivariate normal distribution. Define the ran dom variables V and W by V = a1X +b1Y +c1 and W = a2X +b2Y +c2, where ai, bi and ci are constants for i = 1,2. Argue that (V,W) has a bivariate normal distribution.
1. Problem 12.4 Use the decomposition of the standard bivariate normal density to verify that P(X ≤ a,Y ≤b) can be calculated as 1√2π−∞2 x2 dx(b −μ2)/σ2 −ρx 1 −ρ2 e−1 if the random vector (X,Y) is bivariate normal distributed with parameters(μ1,μ2,σ2 1,σ2 2,ρ). Here, (x)
1. Problem 12.3 Suppose that the probability density function f (x, y) of the random variables X and Y is given by the bivariate standard normal density with parameter ρ. Verify that the probability density function fZ(z) of the ratio Z = X/Y isgiven by the so-called Cauchy density∞−∞|y| f
1. Problem12.2Theratesofreturnontwostocks Aand B haveabivariatenormal distribution with parameters μ1 = 0.08, μ2= 0.12, σ1= 0.05, σ2= 0.15, and ρ =−0.50. What is the probability that the average of the returns on stocks A and B will be larger than 0.11?
1.Problem 12.1 Prove that aX+bY is normally distributed for any constants a and b if (X,Y) has a bivariate normal distribution. How do you calculate P(X >Y)?
1. Problem 11.20 What is the correlation coefficient of the Cartesian coordinates of a point picked at random in the unit circle?
1. Problem11.19 Youroll a pair of dice. What is the correlation coefficient of the high and low points rolled?
1. Problem 11.18 What is the variance of the number of distinct birthdays within a randomly formed group of 100 persons? Hint: define the random variable Xi as1iftheith day is among the 100 birthdays, and as 0 otherwise.
1. Problem11.17Thehypergeometricdistribution describes the probability mass function of the number of red balls drawn when n balls are randomly chosen from an urn containing R red and W white balls. Show that the variance of the number of red balls drawn is given by n R R+W(1 − R R+W)R+W−n
1. Problem 11.16 Let the random variables X1,...,Xn be defined on a common probability space. Prove that nvar(X1 +···+Xn) =i=1 nvar(Xi) + 2 i=1 nj=i+1 cov(Xi, Xj).Next, evaluate var( n i=1 ti Xi) in order to verify that n i=1 nj=1 titjσij ≥ 0 for all real numbers t1,...,tn, where σij =

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Understanding Probability 2nd Edition Henk Tijms - Solutions

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