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Unformatted text preview: One Thousand Exercises in Probability Geoffrey R. Grimmett Statistical Laboratory, Cambridge University David R. Stirzaker Mathematical Institute, Oxford University 18 April 2001 iii Preface This book contains more than 1000 exercises in probability and random processes, together with their solutions. Apart from being a volume of worked exercises in its own right, it is also a solutions manual for exercises and problems appearing in our textbook Probability and Random Processes (3rd edn), Oxford University Press, 2001, henceforth referred to as PRP. These exercises are not merely for drill, but complement and illustrate the text of PRP, or are entertaining, or both. The current volume extends our earlier book Probability and Random Processes: Problems and Solutions, and includes in addition around 400 new problems. Since many exercises have multiple parts, the total number of interrogatives exceeds 3000. Despite being intended in part as a companion to PRP, the present volume is as selfcontained as reasonably possible. Where knowledge of a substantial chunk of bookwork is unavoidable, the reader is provided with a reference to the relevant passage in PRP. Expressions such as ‘clearly’ appear frequently in the solutions. Although we do not use such terms in their Laplacian sense to mean ‘with difficulty’, to call something ‘clear’ is not to imply that explicit verification is necessarily free of tedium. The table of contents reproduces that of PRP; the section and exercise numbers correspond to those of PRP; there are occasional references to examples and equations in PRP. The covered range of topics is broad, beginning with the elementary theory of probability and random variables, and continuing, via chapters on Markov chains and convergence, to extensive sections devoted to stationarity and ergodic theory, renewals, queues, martingales, and diffusions, including an introduction to the pricing of options. Generally speaking, exercises are questions which test knowledge of particular pieces of theory, while problems are less specific in their requirements. There are questions of all standards, the great majority being elementary or of intermediate difficulty. We ourselves have found some of the later ones to be rather tricky, but have refrained from magnifying any difficulty by adding asterisks or equivalent devices. If you are using this book for self-study, our advice would be not to attempt more than a respectable fraction of these at a first read. We pay tribute to all those anonymous pedagogues whose examination papers, work assignments, and textbooks have been so influential in the shaping of this collection. To them and to their successors we wish, in turn, much happy plundering. If you find errors, try to keep them secret, except from us. If you know a better solution to any exercise, we will be happy to substitute it in a later edition. We acknowledge the expertise of Sarah Shea-Simonds in preparing the T EXscript of this volume, and of Andy Burbanks in advising on the front cover design, which depicts a favourite confluence of the authors. Cambridge and Oxford April 2001 G. R. G. D. R. S. v Life is good for only two things, discovering mathematics and teaching it. Sim´eon Poisson In mathematics you don’t understand things, you just get used to them. John von Neumann Probability is the bane of the age. Anthony Powell Casanova’s Chinese Restaurant The traditional professor writes a, says b, and means c; but it should be d. George Po´ lya Contents 1 Introduction Events as sets Probability Conditional probability Independence Completeness and product spaces Worked examples Problems 1 1 2 3 135 135 137 139 4 4 140 141 10 10 11 11 12 151 152 152 152 153 12 154 16 16 17 18 19 19 20 21 22 23 23 158 158 161 162 165 165 167 169 170 171 172 Random variables and their distributions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3 Solutions Events and their probabilities 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2 Questions Random variables The law of averages Discrete and continuous variables Worked examples Random vectors Monte Carlo simulation Problems Discrete random variables 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Probability mass functions Independence Expectation Indicators and matching Examples of discrete variables Dependence Conditional distributions and conditional expectation Sums of random variables Simple random walk Random walk: counting sample paths Problems vii Contents 4 Continuous random variables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 5 29 29 30 30 31 32 33 34 35 36 36 37 38 39 187 188 189 190 191 193 195 199 201 202 204 205 206 209 48 49 50 51 52 52 53 54 55 56 57 57 230 232 234 238 239 241 241 244 247 249 253 254 64 65 66 67 68 69 70 71 72 272 275 276 281 286 287 289 290 293 73 74 74 75 76 297 299 301 303 304 Generating functions and their applications 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 6 Probability density functions Independence Expectation Examples of continuous variables Dependence Conditional distributions and conditional expectation Functions of random variables Sums of random variables Multivariate normal distribution Distributions arising from the normal distribution Sampling from a distribution Coupling and Poisson approximation Geometrical probability Problems Generating functions Some applications Random walk Branching processes Age-dependent branching processes Expectation revisited Characteristic functions Examples of characteristic functions Inversion and continuity theorems Two limit theorems Large deviations Problems Markov chains 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 Markov processes Classification of states Classification of chains Stationary distributions and the limit theorem Reversibility Chains with finitely many states Branching processes revisited Birth processes and the Poisson process Continuous-time Markov chains Uniform semigroups Birth–death processes and imbedding Special processes Spatial Poisson processes Markov chain Monte Carlo Problems viii Contents 7 Convergence of random variables 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 8 85 85 86 88 88 89 89 90 90 91 91 323 323 326 330 331 331 331 332 333 334 336 97 97 98 99 349 350 351 352 99 353 101 101 102 102 103 103 104 355 356 357 359 359 360 361 107 107 108 108 109 109 370 371 373 375 375 376 112 113 113 113 114 114 115 382 384 384 385 386 386 387 Random processes 8.1 8.2 8.3 8.4 8.5 8.6 8.7 9 Introduction Modes of convergence Some ancillary results Laws of large numbers The strong law The law of the iterated logarithm Martingales Martingale convergence theorem Prediction and conditional expectation Uniform integrability Problems Introduction Stationary processes Renewal processes Queues The Wiener process Existence of processes Problems Stationary processes 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Introduction Linear prediction Autocovariances and spectra Stochastic integration and the spectral representation The ergodic theorem Gaussian processes Problems 10 Renewals 10.1 10.2 10.3 10.4 10.5 10.6 The renewal equation Limit theorems Excess life Applications Renewal–reward processes Problems 11 Queues 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Single-server queues M/M/1 M/G/1 G/M/1 G/G/1 Heavy traffic Networks of queues Problems ix Contents 12 Martingales 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 Introduction Martingale differences and Hoeffding’s inequality Crossings and convergence Stopping times Optional stopping The maximal inequality Backward martingales and continuous-time martingales Some examples Problems 118 119 119 120 120 396 398 398 399 400 121 403 121 403 126 127 127 127 127 128 129 129 130 130 411 413 413 413 415 416 417 418 420 420 13 Diffusion processes 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 Introduction Brownian motion Diffusion processes First passage times Barriers Excursions and the Brownian bridge Stochastic calculus The It oˆ integral Itoˆ ’s formula Option pricing Passage probabilities and potentials Problems Bibliography 429 Index 430 x 1 Events and their probabilities 1.2 Exercises. Events as sets 1. Let { Ai : i ∈ I } be a collection of sets. Prove ‘De Morgan’s Laws’†: !" i 2. Ai #c = $ !$ Aci , i i Ai #c = " Aci . i Let A and B belong to some σ -field F. Show that F contains the sets A ∩ B, A \ B, and A △ B. 3. A conventional knock-out tournament (such as that at Wimbledon) begins with 2n competitors and has n rounds. There are no play-offs for the positions 2, 3, . . . , 2n − 1, and the initial table of draws is specified. Give a concise description of the sample space of all possible outcomes. 4. Let F be a σ -field of subsets of " and suppose that B ∈ F. Show that G = { A ∩ B : A ∈ F} is a σ -field of subsets of B. 5. (a) (b) (c) (d) Which of the following are identically true? For those that are not, say when they are true. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∩ C) = (A ∩ B) ∩ C; (A ∪ B) ∩ C = A ∪ (B ∩ C); A \ (B ∩ C) = (A \ B) ∪ (A \ C). 1.3 Exercises. Probability 1 ≤ P(A∩ B) ≤ 1 , 1. Let A and B be events with probabilities P(A) = 34 and P(B) = 13 . Show that 12 3 and give examples to show that both extremes are possible. Find corresponding bounds for P(A ∪ B). 2. A fair coin is tossed repeatedly. Show that, with probability one, a head turns up sooner or later. Show similarly that any given finite sequence of heads and tails occurs eventually with probability one. Explain the connection with Murphy’s Law. 3. Six cups and saucers come in pairs: there are two cups and saucers which are red, two white, and two with stars on. If the cups are placed randomly onto the saucers (one each), find the probability that no cup is upon a saucer of the same pattern. †Augustus De Morgan is well known for having given the first clear statement of the principle of mathematical induction. He applauded probability theory with the words: “The tendency of our study is to substitute the satisfaction of mental exercise for the pernicious enjoyment of an immoral stimulus”. 1 [1.3.4]–[1.4.5] Exercises Events and their probabilities Let A1 , A2 , . . . , An be events where n ≥ 2, and prove that 4. P !" n i=1 Ai # = % i P(Ai ) − % i< j P(Ai ∩ A j ) + % i< j <k P(Ai ∩ A j ∩ Ak ) − · · · + (−1)n+1 P(A1 ∩ A2 ∩ · · · ∩ An ). In each packet of Corn Flakes may be found a plastic bust of one of the last five Vice-Chancellors of Cambridge University, the probability that any given packet contains any specific Vice-Chancellor being 15 , independently of all other packets. Show that the probability that each of the last three Vice-Chancellors is obtained in a bulk purchase of six packets is 1 − 3( 45 )6 + 3( 35 )6 − ( 25 )6 . &'∞ ( 5. Let Ar , r ≥ 1, be events such that P(Ar ) = 1 for all r . Show that P r=1 Ar = 1. 6. You are given that at least one of the events Ar , 1 ≤ r ≤ n, is certain to occur, but certainly no more than two occur. If P(Ar ) = p, and P(Ar ∩ As ) = q, r ̸= s, show that p ≥ 1/n and q ≤ 2/n. 7. You are given that at least one, but no more than three, of the events Ar , 1 ≤ r ≤ n, occur, where n ≥ 3. The probability of at least two occurring is 12 . If P(Ar ) = p, P(Ar ∩ As ) = q, r ̸= s, and P(Ar ∩ As ∩ At ) = x, r < s < t, show that p ≥ 3/(2n), and q ≤ 4/n. 1.4 Exercises. Conditional probability 1. Prove that P(A | B) = P(B | A)P(A)/P(B) whenever P(A)P(B) ̸= 0. Show that, if P(A | B) > P(A), then P(B | A) > P(B). 2. For events A1 , A2 , . . . , An satisfying P(A1 ∩ A2 ∩ · · · ∩ An−1 ) > 0, prove that P(A1 ∩ A2 ∩ · · · ∩ An ) = P(A1 )P(A2 | A1 )P(A3 | A1 ∩ A2 ) · · · P(An | A1 ∩ A2 ∩ · · · ∩ An−1 ). 3. A man possesses five coins, two of which are double-headed, one is double-tailed, and two are normal. He shuts his eyes, picks a coin at random, and tosses it. What is the probability that the lower face of the coin is a head? He opens his eyes and sees that the coin is showing heads; what is the probability that the lower face is a head? He shuts his eyes again, and tosses the coin again. What is the probability that the lower face is a head? He opens his eyes and sees that the coin is showing heads; what is the probability that the lower face is a head? He discards this coin, picks another at random, and tosses it. What is the probability that it shows heads? 4. What do you think of the following ‘proof’ by Lewis Carroll that an urn cannot contain two balls of the same colour? Suppose that the urn contains two balls, each of which is either black or white; thus, in the obvious notation, P(BB) = P(BW) = P(WB) = P(WW) = 14 . We add a black ball, so that P(BBB) = P(BBW) = P(BWB) = P(BWW) = 14 . Next we pick a ball at random; the chance that the ball is black is (using conditional probabilities) 1 · 14 + 23 · 14 + 23 · 14 + 13 · 14 = 23 . However, if there is probability 23 that a ball, chosen randomly from three, is black, then there must be two black and one white, which is to say that originally there was one black and one white ball in the urn. 5. The Monty Hall problem: goats and cars. (a) Cruel fate has made you a contestant in a game show; you have to choose one of three doors. One conceals a new car, two conceal old goats. You 2 Independence Exercises [1.4.6]–[1.5.7] choose, but your chosen door is not opened immediately. Instead, the presenter opens another door to reveal a goat, and he offers you the opportunity to change your choice to the third door (unopened and so far unchosen). Let p be the (conditional) probability that the third door conceals the car. The value of p depends on the presenter’s protocol. Devise protocols to yield the values p = 12 , p = 23 . Show that, for α ∈ [ 12 , 23 ], there exists a protocol such that p = α. Are you well advised to change your choice to the third door? (b) In a variant of this question, the presenter is permitted to open the first door chosen, and to reward you with whatever lies behind. If he chooses to open another door, then this door invariably conceals a goat. Let p be the probability that the unopened door conceals the car, conditional on the presenter having chosen to open a second door. Devise protocols to yield the values p = 0, p = 1, and deduce that, for any α ∈ [0, 1], there exists a protocol with p = α. 6. The prosecutor’s fallacy†. Let G be the event that an accused is guilty, and T the event that some testimony is true. Some lawyers have argued on the assumption that P(G | T ) = P(T | G). Show that this holds if and only if P(G) = P(T ). 7. Urns. There are n urns of which the r th contains r − 1 red balls and n − r magenta balls. You pick an urn at random and remove two balls at random without replacement. Find the probability that: (a) the second ball is magenta; (b) the second ball is magenta, given that the first is magenta. 1.5 Exercises. Independence 1. Let A and B be independent events; show that Ac , B are independent, and deduce that Ac , B c are independent. 2. We roll a die n times. Let Ai j be the event that the i th and j th rolls produce the same number. Show that the events { Ai j : 1 ≤ i < j ≤ n} are pairwise independent but not independent. 3. A fair coin is tossed repeatedly. Show that the following two statements are equivalent: (a) the outcomes of different tosses are independent, (b) for any given finite sequence of heads and tails, the chance of this sequence occurring in the first m tosses is 2−m , where m is the length of the sequence. 4. Let " = {1, 2, . . . , p} where p is prime, F be the set of all subsets of ", and P(A) = | A|/ p for all A ∈ F. Show that, if A and B are independent events, then at least one of A and B is either ∅ or ". 5. Show that the conditional independence of A and B given C neither implies, nor is implied by, the independence of A and B. For which events C is it the case that, for all A and B, the events A and B are independent if and only if they are conditionally independent given C? 6. Safe or sorry? Some form of prophylaxis is said to be 90 per cent effective at prevention during one years treatment. If the degrees of effectiveness in different years are independent, show that the treatment is more likely than not to fail within 7 years. 7. Families. Jane has three children, each of which is equally likely to be a boy or a girl independently of the others. Define the events: A = {all the children are of the same sex}, B = {there is at most one boy}, C = {the family includes a boy and a girl}. †The prosecution made this error in the famous Dreyfus case of 1894. 3 [1.5.8]–[1.8.3] (a) (b) (c) (d) Exercises Events and their probabilities Show that A is independent of B, and that B is independent of C. Is A independent of C? Do these results hold if boys and girls are not equally likely? Do these results hold if Jane has four children? 8. Galton’s paradox. You flip three fair coins. At least two are alike, and it is an evens chance that the third is a head or a tail. Therefore P(all alike) = 21 . Do you agree? 9. Two fair dice are rolled. Show that the event that their sum is 7 is independent of the score shown by the first die. 1.7 Exercises. Worked examples 1. There are two roads from A to B and two roads from B to C. Each of the four roads is blocked by snow with probability p, independently of the others. Find the probability that there is an open road from A to B given that there is no open route from A to C. If, in addition, there is a direct road from A to C, this road being blocked with probability p independently of the others, find the required conditional probability. 2. Calculate the probability that a hand of 13 cards dealt from a normal shuffled pack of 52 contains exactly two kings and one ace. What is the probability that it contains exactly one ace given that it contains exactly two kings? 3. A symmetric random walk takes place on the integers 0, 1, 2, . . . , N with absorbing barriers at 0 and N , starting at k. Show that the probability that the walk is never absorbed is zero. 4. The so-called ‘sure thing principle’ asserts that if you prefer x to y given C, and also prefer x to y given C c , then you surely prefer x to y. Agreed? 5. A pack contains m cards, labelled 1, 2, . . . , m. The cards are dealt out in a random order, one by one. Given that the label of the kth card dealt is the largest of the first k cards dealt, what is the probability that it is also the largest in the pack? 1.8 Problems 1. (a) (b) (c) (d) A traditional fair die is thrown twice. What is the probability that: a six turns up exactly once? both numbers are odd? the sum of the scores is 4? the sum of the scores is divisible by 3? 2. (a) (b) (c) (d) A fair coin is thrown repeatedly. What is the probability that on the nth throw: a head appears for the first time? the numbers of heads and tails to date are equal? exactly two heads have appeared altogether to date? at least two heads have appeared to date? 3. Let F and G be σ -fields of subsets of ". (a) Use elementary set operations to 'show that F is closed under countable intersections; that is, if A1 , A2 , . . . are in F, then so is i Ai . (b) Let H = F ∩ G be the collection of subsets of " lying in both F and G. Show that H is a σ -field. (c) Show that F ∪ G, the collection of subsets of " lying in either F or G, is not necessarily a σ -field. 4 Problems Exercises [1.8.4]–[1.8.14] 4. Describe the underlying probability spaces for the following experiments: (a) a biased coin is tossed three times; (b) two balls are drawn without replacement from an urn which originally contained two ultramarine and two vermilion balls; (c) a biased coin is tossed repeatedly until a head turns up. 5. Show that the probability that exactly one of the events A and B occurs is P(A) + P(B) − 2P(A ∩ B). Prove that P(A ∪ B ∪ C) = 1 − P(Ac | B c ∩ C c )P(B c | C c )P(C c ). 6. 7. (a) If A is independent of itself, show that P(A) is 0 or 1. (b) If P(A) is 0 or 1, show that A is independent of all events B. 8. Let F be a σ -field of subsets of ", and suppose P : F → [0, 1] satisfies: (i) P(") = 1, and (ii) P i...
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