![6 SOME BACKGROUND ON PROBABILITY
which justifies (1.2).
If the probability of B is unaffected by the prior occurrence of 𝐴, then we say that
𝐴 and 𝐵 are independent or that
P(𝐵∣𝐴) = P(𝐵),
which from above implies that
P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵).
Conversely, if P(𝐵∣𝐴) = P(𝐵), then 𝐴 and 𝐵 are independent events. Again this
result can be extended to 3 or more independent events.
Example 1.3 Let 𝐴 and 𝐵 be independent events with P(𝐴) = 1
4
and P(𝐵) = 2
3
. Calculate
the following probabilities: (a) P(𝐴 ∩ 𝐵); (b) P(𝐴 ∩ 𝐵𝑐
); (c) P(𝐴𝑐
∩ 𝐵𝑐
); (d) P(𝐴𝑐
∩ 𝐵);
(e) P((𝐴 ∪ 𝐵)𝑐
).
Since the events are independent, then P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵). Hence
(a) P(𝐴 ∩ 𝐵) = 1
4
⋅ 2
3
= 1
6
.
(b) The independence 𝐴 and 𝐵𝑐
follows by eliminating P(𝐴 ∩ 𝐵) between the equations
P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵) = P(𝐴)[1 − P(𝐵𝑐
)]
and
P(𝐴) = P[(𝐴 ∩ 𝐵𝑐
) ∪ (𝐴 ∩ 𝐵)] = P(𝐴 ∩ 𝐵𝑐
) + P(𝐴 ∩ 𝐵).
Hence
P(𝐴 ∩ 𝐵𝑐
) = P(𝐴)P(𝐵𝑐
) = P(𝐴)[1 − P(𝐵)] = 1
4
(1 − 2
3
) = 1
12
.
(c) Since 𝐴𝑐
and 𝐵𝑐
are independent events,
P(𝐴𝑐
∩ 𝐵𝑐
) = P(𝐴𝑐
)P(𝐵𝑐
) = [1 − P(𝐴)][1 − P(𝐵)] = 3
4
⋅ 1
3
= 1
4
.
(d) Since 𝐴𝑐
and 𝐵 are independent events, P(𝐴𝑐
∩ 𝐵) = P(𝐴𝑐
)P(𝐵) = [1 − 1
4
]2
3
= 1
2
.
(e) P((𝐴 ∪ 𝐵)𝑐
) = 1 − P(𝐴 ∪ 𝐵) = 1 − P(𝐴) − P(𝐵) + P(𝐴 ∩ 𝐵) by (1.1). Hence
P((𝐴 ∪ 𝐵)𝑐
) = 1 − P(𝐴) − P(𝐵) + P(𝐴)P(𝐵) = 1 − 1
4
− 2
3
+ 1
6
= 1
4
.
Example 1.4. For three events 𝐴, 𝐵, and 𝐶, show that
P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶),
where P(𝐶) > 0.
By using (1.2) and viewing 𝐴 ∩ 𝐵 ∩ 𝐶 as (𝐴 ∩ 𝐵) ∩ 𝐶 or 𝐴 ∩ (𝐵 ∩ 𝐶),
P(𝐴 ∩ 𝐵 ∩ 𝐶) = P(𝐴 ∩ 𝐵∣𝐶)P(𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵 ∩ 𝐶).
Hence
P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶)
P(𝐵 ∩ 𝐶)
P(𝐶)
= P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶)
by (1.2) again.
A result known as the law of total probability or the partition theorem will](https://image.slidesharecdn.com/63744-250319184743-c792fb86/85/Stochastic-Processes-An-Introduction-2nd-Edition-Peter-Watts-Jones-22-320.jpg)
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- 10. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130920 International Standard Book Number-13: 978-1-4398-7122-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 11. Contents Preface ix 1 Some Background on Probability 1 1.1 Introduction 1 1.2 Probability 1 1.3 Conditional probability and independence 5 1.4 Discrete random variables 7 1.5 Continuous random variables 9 1.6 Mean and variance 10 1.7 Some standard discrete probability distributions 12 1.8 Some standard continuous probability distributions 14 1.9 Generating functions 17 1.10 Conditional expectation 21 1.11 Problems 24 2 Some Gambling Problems 29 2.1 Gambler’s ruin 29 2.2 Probability of ruin 29 2.3 Some numerical simulations 33 2.4 Duration of the game 34 2.5 Some variations of gambler’s ruin 38 2.5.1 The infinitely rich opponent 38 2.5.2 The generous opponent 38 2.5.3 Changing the stakes 39 2.6 Problems 39 3 Random Walks 45 3.1 Introduction 45 3.2 Unrestricted random walks 46 3.3 The general probability distribution of a walk 48 3.4 First returns of the symmetric random walk 50 3.5 Problems 52 4 Markov Chains 59 4.1 States and transitions 59 4.2 Transition probabilities 60 v
- 12. vi CONTENTS 4.3 General two-state Markov chains 64 4.4 Powers of the general transition matrix 66 4.5 Gambler’s ruin as a Markov chain 73 4.6 Classification of states 76 4.7 Classification of chains 83 4.8 Problems 86 5 Poisson Processes 93 5.1 Introduction 93 5.2 The Poisson process 93 5.3 Partition theorem approach 96 5.4 Iterative method 97 5.5 The generating function 98 5.6 Variance in terms of the probability generating function 100 5.7 Arrival times 101 5.8 Summary of the Poisson process 103 5.9 Problems 104 6 Birth and Death Processes 107 6.1 Introduction 107 6.2 The birth process 107 6.3 Birth process: Generating function equation 110 6.4 The death process 112 6.5 The combined birth and death process 115 6.6 General population processes 119 6.7 Problems 122 7 Queues 131 7.1 Introduction 131 7.2 The single-server queue 132 7.3 The stationary process 134 7.4 Queues with multiple servers 140 7.5 Queues with fixed service times 144 7.6 Classification of queues 147 7.7 A general approach to the 𝑀(𝜆)/𝐺/1 queue 147 7.8 Problems 151 8 Reliability and Renewal 157 8.1 Introduction 157 8.2 The reliability function 157 8.3 Exponential distribution and reliability 159 8.4 Mean time to failure 160 8.5 Reliability of series and parallel systems 161 8.6 Renewal processes 163 8.7 Expected number of renewals 165
- 13. CONTENTS vii 8.8 Problems 167 9 Branching and Other Random Processes 171 9.1 Introduction 171 9.2 Generational growth 171 9.3 Mean and variance 174 9.4 Probability of extinction 176 9.5 Branching processes and martingales 179 9.6 Stopping rules 182 9.7 The simple epidemic 184 9.8 An iterative solution scheme for the simple epidemic 186 9.9 Problems 188 10 Computer Simulations and Projects 195 Answers and Comments on End-of-Chapter Problems 203 Appendix 211 References and Further Reading 215 Index 217
- 15. Preface This textbook was developed from a course in stochastic processes given by the au- thors over many years to second-year students studying Mathematics or Statistics at Keele University. At Keele the majority of students take degrees in Mathematics or Statistics jointly with another subject, which may be from the sciences, social sci- ences or humanities. For this reason the course has been constructed to appeal to students with varied academic interests, and this is reflected in the book by including applications and examples that students can quickly understand and relate to. In par- ticular, in the earlier chapters, the classical gambler’s ruin problem and its variants are modeled in a number of ways to illustrate simple random processes. Specialized applications have been avoided to accord with our view that students have enough to contend with in the mathematics required in stochastic processes. Topics can be selected from Chapters 2 to 9 for a one-semester course or mod- ule in random processes. It is assumed that readers have already encountered the usual first-year courses in calculus and matrix algebra and have taken a first course in probability; nevertheless, a revision of relevant basic probability is included for reference in Chapter 1. Some of the easier material on discrete random processes is included in Chapters 2, 3, and 4, which cover some simple gambling problems, ran- dom walks, and Markov chains. Random processes continuous in time are developed in Chapters 5 and 6. These include Poisson, birth and death processes, and general population models. Continuous time models include queues in Chapter 7, which has an extended discussion on the analysis of associated stationary processes. The book ends with two chapters on reliability and other random processes, the latter including branching processes, martingales, and a simple epidemic. An appendix contains key mathematical results for reference. There are over 50 worked examples in the text and 205 end-of-chapter problems with hints and answers listed at the end of the book. Mathematica𝑇 𝑀 is a mathematical software package able to carry out complex symbolic mathematical as well as numerical computations. It has become an integral part of many degree courses in Mathematics or Statistics. The software has been used throughout the book to solve both theoretical and numerical examples and to produce many of the graphs. R is a statistical computing and graphics package which is available free of charge, and can be downloaded from: http://www.r-project.org ix
- 16. x PREFACE Like R, S-PLUS (not freeware) is derived from the S language, and hence users of these packages will be able to apply them to the solution of numerical projects, including those involving matrix algebra presented in the text. Mathematica code has been applied to all the projects listed by chapters in Chapter 10, and R code to some as appropriate. All the Mathematica and R programs can be found on the Keele University Web site: http://www.scm.keele.ac.uk/books/stochastic processes/ Not every topic in the book is included, but the programs, which generally use standard commands, are intended to be flexible in that inputs, parameters, data, etc., can be varied by the user. Graphs and computations can often add insight into what might otherwise be viewed as rather mechanical analysis. In addition, more compli- cated examples, which might be beyond hand calculations, can be attempted. We are grateful to staff of the School of Computing and Mathematics, Keele Uni- versity, for help in designing the associated Web site. Finally, we would like to thank the many students at Keele over many years who have helped to develop this book, and to the interest shown by users of the first edition in helping us to refine and update this second edition. Peter W. Jones Peter Smith Keele University
- 17. CHAPTER 1 Some Background on Probability 1.1 Introduction We shall be concerned with the modeling and analysis of random experiments us- ing the theory of probability. The outcome of such an experiment is the result of a stochastic or random process. In particular we shall be interested in the way in which the results or outcomes vary or evolve over time. An experiment or trial is any sit- uation where an outcome is observed. In many of the applications considered, these outcomes will be numerical, sometimes in the form of counts or enumerations. The experiment is random if the outcome is not predictable or is uncertain. At first we are going to be concerned with simple mechanisms for creating random outcomes, namely games of chance. One recurring theme initially will be the study of the classical problem known as gambler’s ruin. We will then move on to applications of probability to modeling in, for example, engineering, medicine, and biology. We make the assumption that the reader is familiar with the basic theory of probability. This background will however be reinforced by the brief review of these concepts which will form the main part of this chapter. 1.2 Probability In random experiments, the list of all possible outcomes is termed the sample space, denoted by 𝑆. This list consists of individual outcomes or elements. These elements have the properties that they are mutually exclusive and that they are exhaustive. Mutually exclusive means that two or more outcomes cannot occur simultaneously: exhaustive means that all possible outcomes are in the list. Thus each time the exper- iment is carried out one of the outcomes in 𝑆 must occur. A collection of elements of 𝑆 is called an event: these are usually denoted by capital letters, 𝐴, 𝐵, etc. We denote by P(𝐴) the probability that the event 𝐴 will occur at each repetition of the random experiment. Remember that 𝐴 is said to have occurred if one element making up 𝐴 has occurred. In order to calculate or estimate the probability of an event 𝐴 there are two possibilities. In one approach an experiment can be performed a large number of times, and P(𝐴) can be approximated by the relative frequency with which 𝐴 occurs. In order to analyze random experiments we make the assumption that the conditions surrounding the trials remain the same, and are independent of one another. We hope 1
- 18. 2 SOME BACKGROUND ON PROBABILITY that some regularity or settling down of the outcome is apparent. The ratio the number of times a particular event 𝐴 occurs total number of trials is known as the relative frequency of the event, and the number to which it ap- pears to converge as the number of trials increases is known as the probability of an outcome within 𝐴. Where we have a finite sample space it might be reasonable to assume that the outcomes of an experiment are equally likely to occur as in the case, for example, in rolling a fair die or spinning an unbiased coin. In this case the probability of 𝐴 is given by P(𝐴) = number of elements of 𝑆 where 𝐴 occurs number of elements in 𝑆 . There are, of course, many ‘experiments’ which are not repeatable. Horse races are only run once, and the probability of a particular horse winning a particular race may not be calculated by relative frequency. However, a punter may form a view about the horse based on other factors which may be repeated over a series of races. The past form of the horse, the form of other horses in the race, the state of the course, the record of the jockey, etc., may all be taken into account in determining the probability of a win. This leads to a view of probability as a ‘degree of belief’ about uncertain outcomes. The odds placed by bookmakers on the horses in a race reflect how punters place their bets on the race. The odds are also set so that the bookmakers expect to make a profit. It is convenient to use set notation when deriving probabilities of events. This leads to 𝑆 being termed the universal set, the set of all outcomes: an event 𝐴 is a subset of 𝑆. This also helps with the construction of more complex events in terms of the unions and intersections of several events. The Venn diagrams shown in Figure 1.1 represent the main set operations of union (∪), intersection (∩), and complement (𝐴𝑐 ) which are required in probability. ∙ Union. The union of two sets 𝐴 and 𝐵 is the set of all elements which belong to 𝐴, or to 𝐵, or to both. It can be written formally as 𝐴 ∪ 𝐵 = {𝑥∣𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 or both}. ∙ Intersection. The intersection of two sets 𝐴 and 𝐵 is the set 𝐴∩𝐵 which contains all elements common to both 𝐴 and 𝐵. It can be written as 𝐴 ∩ 𝐵 = {𝑥∣𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵}. ∙ Complement. The complement 𝐴𝑐 of a set 𝐴 is the set of all elements which belong to the universal set 𝑆 but do not belong to 𝐴. It can be written as 𝐴𝑐 = {𝑥 ∕∈ 𝐴}. So, for example, in an experiment in which we are interested in two events 𝐴 and 𝐵, then 𝐴𝑐 ∩ 𝐵 may be interpreted as ‘only 𝐵’, being the intersection of the complement of 𝐴 and 𝐵 (see Figure 1.1d): this is alternatively expressed in the difference notation 𝐵∖𝐴 meaning 𝐵 but not 𝐴. We denote by 𝜙 the empty set, that is the set which contains no elements. Note that 𝑆𝑐 = 𝜙. Two events 𝐴 and 𝐵 are
- 19. PROBABILITY 3 U U U A A A B B (a) (b) (c) U (d) B A Figure 1.1 (a) the union 𝐴 ∪ 𝐵 of 𝐴 and 𝐵; (b) the intersection 𝐴 ∩ 𝐵 of 𝐴 and 𝐵; (c) the complement 𝐴𝑐 of 𝐴: 𝑆 is the universal set; (d) 𝐴𝑐 ∩ 𝐵 or 𝐵∖𝐴 said to be mutually exclusive if 𝐴 and 𝐵 have no events in common 𝐴 and 𝐵 so that 𝐴 ∩ 𝐵 = 𝜙, the empty set: in set terminology 𝐴 and 𝐵 are said to be disjoint sets. The probability of any event satisfies however calculated the three axioms ∙ Axiom 1: 0 ≤ P(𝐴) ≤ 1 for every event 𝐴 ∙ Axiom 2: P(𝑆) = 1 ∙ Axiom 3: P(𝐴∪𝐵) = P(𝐴)+P(𝐵) if 𝐴 and 𝐵 are mutually exclusive (𝐴∩𝐵 = 𝜙) Axiom 3 may be extended to more than two mutually exclusive events, say 𝑘 of them represented by 𝐴1, 𝐴2, . . . , 𝐴𝑘 where 𝐴𝑖 ∩ 𝐴𝑗 = 𝜙 for all 𝑖 ∕= 𝑗. This is called a partition of 𝑆 if ∙ (a) 𝐴𝑖 ∩ 𝐴𝑗 = 𝜙 for all 𝑖 ∕= 𝑗, ∙ (b) 𝑘 ∪ 𝑖=1 𝐴𝑖 = 𝐴1 ∪ 𝐴2 ∪ . . . ∪ 𝐴𝑘 = 𝑆, ∙ (c) P(𝐴𝑖) > 0. In this definition, (a) states that the events are mutually exclusive, (b) that every event in 𝑆 occurs in one of the events 𝐴𝑖, and (c) implies that there is a nonzero probability that any 𝐴𝑖 occurs. It follows that 1 = P(𝑆) = P(𝐴1 ∪ 𝐴2 ∪ ⋅ ⋅ ⋅ ∪ 𝐴𝑘) = 𝑘 ∑ 𝑖=1 P(𝐴𝑖). Theorem ∙ (a) P(𝐴𝑐 ) = 1 − P(𝐴); ∙ (b) P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵). (a) Axiom 3 may be combined with Axiom 2 to give P(𝐴𝑐 ), the probability that the
- 20. 4 SOME BACKGROUND ON PROBABILITY complement 𝐴𝑐 occurs, by noting that 𝑆 = 𝐴 ∪ 𝐴𝑐 . This is a partition of 𝑆 into the mutually exclusive exhaustive events 𝐴 and 𝐴𝑐 . Thus 1 = P(𝑆) = P(𝐴 ∪ 𝐴𝑐 ) = P(𝐴) + P(𝐴𝑐 ), giving P(𝐴𝑐 ) = 1 − P(𝐴). (b) For any sets 𝐴 and 𝐵 𝐴 ∪ 𝐵 = 𝐴 ∪ (𝐵 ∩ 𝐴𝑐 ), and 𝐵 = (∩𝐵) ∪ (𝐵 ∩ 𝐴𝑐 ), in which 𝐴 and 𝐵 ∩ 𝐴𝑐 are disjoint sets, and 𝐴 ∩ 𝐵 and 𝐵 ∩ 𝐴𝑐 are disjoint sets. Therefore, by Axiom 3, P(𝑎 ∪ 𝐵) = P(𝐴) + P(𝐵 ∩ 𝐴𝑐 ), and P(𝐵) = P(𝐴 ∩ 𝐵) + P(𝐵 ∩ 𝐴𝑐 ). Elimination of P(𝐵 ∩ 𝐴𝑐 ) between these equations leads to P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵) (1.1) as required. Example 1.1. Two distinguishable fair dice 𝑎 and 𝑏 are rolled and the values on the uppermost faces noted. What are the elements of the sample space? What is the probability that the sum of the face values of the two dice is 7? What is the probability that at least one 5 appears? We distinguish first the outcome of each die so that there are 6 × 6 = 36 possible outcomes for the pair. The sample space has 36 elements of the form (𝑖, 𝑗) where 𝑖 and 𝑗 take all integer values 1, 2, 3, 4, 5, 6, and 𝑖 is the outcome of die 𝑎 and 𝑗 is the outcome of 𝑏. The full list is 𝑆 = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }, and they are all assumed to be equally likely since the dice are fair. If 𝐴1 is the event that the sum of the dice is 7, then from the list, 𝐴1 = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)} which occurs for 6 elements out of 36. Hence 𝑃(𝐴1) = 6 36 = 1 6 . The event that at least one 5 appears is the list 𝐴2 = {(1, 5), (2, 5), (3, 5), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 5)},
- 21. CONDITIONAL PROBABILITY AND INDEPENDENCE 5 which has 11 elements. Hence 𝑃(𝐴2) = 11 36 . Example 1.2 From a well-shuffled pack of 52 playing cards a single card is randomly drawn. Find the probability that it is a heart or an ace. Let 𝐴 be the event that the card is an ace, and 𝐵 the event that it is a heart. The event 𝐴∩𝐵 is the ace of hearts. We require the probability that it is an ace or a heart, which is P(𝐴 ∪ 𝐵). However, since one of the aces is a heart the events are not mutually exclusive. Hence, we must use eqn (1.1). It follows that the probability that an ace is drawn is P(𝐴) = 4/52, the probability that a heart is drawn is P(𝐵) = 13/52 = 1/4, the probability that the ace of hearts is drawn is P(𝐴 ∩ 𝐵) = 1/52. From (1.1) P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵) = 4 52 + 1 4 − 1 52 = 16 52 = 4 13 . This example illustrates events which are not mutually exclusive. The result could also be obtained directly by noting that 16 of the 52 cards are either hearts or aces. In passing note that 𝐴 ∩ 𝐵𝑐 is the set of aces excluding the ace of hearts, whilst 𝐴𝑐 ∩ 𝐵 is the heart suit excluding the ace of hearts. Hence P(𝐴 ∩ 𝐵𝑐 ) = 3 52 , P(𝐴𝑐 ∩ 𝐵) = 12 52 = 3 13 . 1.3 Conditional probability and independence If the occurrence of an event 𝐵 is affected by the occurrence of another event 𝐴 then we say that 𝐴 and 𝐵 are dependent events. We might be interested in a random ex- periment with which 𝐴 and 𝐵 are associated. When the experiment is performed, it is known that event 𝐴 has occurred. Does this affect the probability of 𝐵? This prob- ability of 𝐵 now becomes the conditional probability of 𝐵 given 𝐴, which is now written as P(𝐵∣𝐴). Usually this will be distinct from the probability P(𝐵). Strictly speaking, this probability is conditional since we must assume that 𝐵 is conditional on the sample space occurring, but it is implicit in P(𝐵). On the other hand the con- ditional probability of 𝐵 is restricted to that part of the sample space where 𝐴 has occurred. This conditional probability is defined as P(𝐵∣𝐴) = P(𝐴 ∩ 𝐵) P(𝐴) , P(𝐴) > 0. (1.2) In terms of counting, suppose that an experiment is repeated N times, of which 𝐴 occurs N(𝐴) times, and 𝐴 given by 𝐵 occurs N(𝐵 ∩ 𝐴) times. The proportion of times that 𝐵 occurs is N(𝐵 ∩ 𝐴) N(𝐴) = N(𝐴 ∩ 𝐵) N N N(𝐵) ,
- 22. 6 SOME BACKGROUND ON PROBABILITY which justifies (1.2). If the probability of B is unaffected by the prior occurrence of 𝐴, then we say that 𝐴 and 𝐵 are independent or that P(𝐵∣𝐴) = P(𝐵), which from above implies that P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵). Conversely, if P(𝐵∣𝐴) = P(𝐵), then 𝐴 and 𝐵 are independent events. Again this result can be extended to 3 or more independent events. Example 1.3 Let 𝐴 and 𝐵 be independent events with P(𝐴) = 1 4 and P(𝐵) = 2 3 . Calculate the following probabilities: (a) P(𝐴 ∩ 𝐵); (b) P(𝐴 ∩ 𝐵𝑐 ); (c) P(𝐴𝑐 ∩ 𝐵𝑐 ); (d) P(𝐴𝑐 ∩ 𝐵); (e) P((𝐴 ∪ 𝐵)𝑐 ). Since the events are independent, then P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵). Hence (a) P(𝐴 ∩ 𝐵) = 1 4 ⋅ 2 3 = 1 6 . (b) The independence 𝐴 and 𝐵𝑐 follows by eliminating P(𝐴 ∩ 𝐵) between the equations P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵) = P(𝐴)[1 − P(𝐵𝑐 )] and P(𝐴) = P[(𝐴 ∩ 𝐵𝑐 ) ∪ (𝐴 ∩ 𝐵)] = P(𝐴 ∩ 𝐵𝑐 ) + P(𝐴 ∩ 𝐵). Hence P(𝐴 ∩ 𝐵𝑐 ) = P(𝐴)P(𝐵𝑐 ) = P(𝐴)[1 − P(𝐵)] = 1 4 (1 − 2 3 ) = 1 12 . (c) Since 𝐴𝑐 and 𝐵𝑐 are independent events, P(𝐴𝑐 ∩ 𝐵𝑐 ) = P(𝐴𝑐 )P(𝐵𝑐 ) = [1 − P(𝐴)][1 − P(𝐵)] = 3 4 ⋅ 1 3 = 1 4 . (d) Since 𝐴𝑐 and 𝐵 are independent events, P(𝐴𝑐 ∩ 𝐵) = P(𝐴𝑐 )P(𝐵) = [1 − 1 4 ]2 3 = 1 2 . (e) P((𝐴 ∪ 𝐵)𝑐 ) = 1 − P(𝐴 ∪ 𝐵) = 1 − P(𝐴) − P(𝐵) + P(𝐴 ∩ 𝐵) by (1.1). Hence P((𝐴 ∪ 𝐵)𝑐 ) = 1 − P(𝐴) − P(𝐵) + P(𝐴)P(𝐵) = 1 − 1 4 − 2 3 + 1 6 = 1 4 . Example 1.4. For three events 𝐴, 𝐵, and 𝐶, show that P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶), where P(𝐶) > 0. By using (1.2) and viewing 𝐴 ∩ 𝐵 ∩ 𝐶 as (𝐴 ∩ 𝐵) ∩ 𝐶 or 𝐴 ∩ (𝐵 ∩ 𝐶), P(𝐴 ∩ 𝐵 ∩ 𝐶) = P(𝐴 ∩ 𝐵∣𝐶)P(𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵 ∩ 𝐶). Hence P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶) P(𝐵 ∩ 𝐶) P(𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶) by (1.2) again. A result known as the law of total probability or the partition theorem will
- 23. DISCRETE RANDOM VARIABLES 7 A1 A2 A5 A4 A3 S Figure 1.2 Schematic set view of a partition of 𝑆 into 5 events 𝐴1, . . . , 𝐴5. be used extensively later, for example, in the discrete gambler’s ruin problem (Sec- tion 2.1) and the Poisson process (Section 5.2). Suppose that 𝐴1, 𝐴2, . . . , 𝐴𝑘 repre- sents a partition of 𝑆 into 𝑘 mutually exclusive events in which, interpreted as sets, the sets fill the space 𝑆 but with none of the sets overlapping. Figure 1.2 shows such a scheme. When a random experiment takes place one and only one of the events can take place. Suppose that 𝐵 is another event associated with the same random experiment (Fig- ure 1.2). Then 𝐵 must be made up of the sum of the intersections of 𝐵 with each of the events in the partition. Some of these will be empty but this does not matter. We can say that 𝐵 is the union of the intersections of 𝐵 with each 𝐴𝑖. Thus 𝐵 = 𝑘 ∪ 𝑖=1 𝐵 ∩ 𝐴𝑖, but the significant point is that any pair of these events is mutually exclusive. It follows that P(𝐵) = 𝑘 ∑ 𝑖=1 P(𝐵 ∩ 𝐴𝑖). (1.3) Since, from equation (1.2), P(𝐵 ∩ 𝐴𝑖) = P(𝐵∣𝐴𝑖)P(𝐴𝑖), equation (1.3) can be expressed as P(𝐵) = 𝑘 ∑ 𝑖=1 P(𝐵∣𝐴𝑖)P(𝐴𝑖), which is the law of total probability or the partition theorem. 1.4 Discrete random variables In most of the applications considered in this text, the outcome of the experiment will be numerical. A random variable usually denoted by the capital letters 𝑋, 𝑌 , or 𝑍, say, is a numerical value associated with the outcome of a random experiment.
- 24. 8 SOME BACKGROUND ON PROBABILITY If 𝑠 is an element of the original sample space 𝑆, which may be numerical or sym- bolic, then 𝑋(𝑠) is a real number associated with 𝑠. The same experiment, of course, may generate several random variables. Each of these random variables will, in turn, have sample spaces whose elements are usually denoted by lower case letters such as 𝑥1, 𝑥2, 𝑥3, . . . for the random variable 𝑋. We are now interested in assigning proba- bilities to events such as P(𝑋 = 𝑥1), the probability that the random variable 𝑋 is 𝑥1 and P(𝑋 ≤ 𝑥2), the probability that the random variable is less than or equal to 𝑥2. If the sample space is finite or countably infinite on the integers (that is, the ele- ments 𝑥0, 𝑥1, 𝑥2, . . . can be counted against integers, say 0, 1, 2, . . .) then we say that the random variable is discrete. Technically, the set {𝑥𝑖} will be a countable subset 𝒱, say, of the real numbers ℛ. We can represent the {𝑥𝑖} generically by the variable 𝑥 with 𝑥 ∈ 𝒱. For example, 𝒱 could be the set {0, 1 2 , 1, 3 2 , 2, 5 2 , 3, . . .}. In many cases 𝒱 consists simply of the integers or a subset of the integers, such as 𝒱 = {0, 1} or 𝒱 = {0, 1, 2, 3, . . .}. In the random walks of Chapter 3, however, 𝒱 may contain all the positive and neg- ative integers . . . − 3, −2, −1, 0, 1, 2, 3, . . .. In these integer cases we can put 𝑥𝑖 = 𝑖. The function 𝑝(𝑥𝑖) = P(𝑋 = 𝑥𝑖) is known as the probability mass function. The pairs {𝑥𝑖, 𝑝(𝑥𝑖)} for all 𝑖 in the sample space define the probability distribution of the random variable 𝑋. If 𝑥𝑖 = 𝑖, which occurs frequently in applications, then 𝑝(𝑥𝑖) = 𝑝(𝑖) is replaced by 𝑝𝑖. Since the 𝑥 values are mutually exclusive and exhaustive then it follows that ∙ (a) 0 ≤ 𝑝(𝑥𝑖) ≤ 1 for all 𝑖, ∙ (b) ∞ ∑ 𝑖=0 𝑝(𝑥𝑖) = 1, or in generic form ∑ 𝑥∈𝒱 𝑝(𝑥) = 1, ∙ (c) P(𝑋 ≤ 𝑥𝑘) = 𝑘 ∑ 𝑖=0 𝑝(𝑥𝑖), which is known as the distribution function. Example 1.5. A fair die is rolled until the first 6 appears face up. Find the probability that the first 6 appears at the 𝑛-th throw. Let the random variable 𝑁 be the number of throws until the first 6 appears face up. This is an example of a discrete random variable 𝑁 with an infinite number of possible outcomes {1, 2, 3, . . .} . The probability of a 6 appearing for any throw is 1 6 and of any other number appearing is 5 6 .
- 25. CONTINUOUS RANDOM VARIABLES 9 Hence the probability of 𝑛 − 1 numbers other than 6 appearing followed by a 6 is P(𝑁 = 𝑛) = (5 6 )𝑛−1 (1 6 ) = 5𝑛−1 6𝑛 , which is the probability mass function for this random variable. 1.5 Continuous random variables In many applications the discrete random variable, which for example might take the integer values 1, 2, . . ., is inappropriate for problems where the random variable can take any real value in an interval. For example, the random variable 𝑇 could be the time measured from time 𝑡 = 0 until a light bulb fails. This could be any value 𝑡 ≥ 0. In this case 𝑇 is called a continuous random variable. Generally, if 𝑋 is a continuous random variable there are mathematical difficulties in defining the event 𝑋 = 𝑥: the probability is usually defined to be zero. Probabilities for continuous random variables may only be defined over intervals of values as, for example, in P(𝑥1 < 𝑋 < 𝑥2). We define a probability density function (pdf) 𝑓(𝑥) over −∞ < 𝑥 < ∞ which has the properties: ∙ (a) 𝑓(𝑥) ≥ 0, (−∞ < 𝑥 < ∞); ∙ (b) P(𝑥1 ≤ 𝑋 ≤ 𝑥2) = ∫ 𝑥2 𝑥1 𝑓(𝑥)𝑑𝑥 for any 𝑥1, 𝑥2 such that −∞ < 𝑥1 < 𝑥2 < ∞; ∙ (c) ∫ ∞ −∞ 𝑓(𝑥)𝑑𝑥 = 1. A possible graph of a density function 𝑓(𝑥) is shown in Figure 1.3. By (a) above the curve must remain nonnegative, by (b) the probability that 𝑋 lies between 𝑥1 f(x) x x1 x2 Figure 1.3 A probability density function. and 𝑥2 is the shaded area, and by (c) the total area under the curve must be 1 since P(−∞ < 𝑋 < ∞) = 1. We define the (cumulative) distribution function (cdf) 𝐹(𝑥) as the probability
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- 29. T h e P r o j e c t G u t e n b e r g e B o o k o f P a u l i n e e t P a s c a l B r u n o
- 30. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Pauline et Pascal Bruno Author: Alexandre Dumas Release date: August 28, 2023 [eBook #71510] Language: French Original publication: Paris: Michel Lévy frères, 1848 Credits: Claudine Corbasson and the online Distributed Proofreaders Canada team at http://www.pgdpcanada.net (This file was produced from images generously made available by The Internet Archive/Canadian Libraries.) *** START OF THE PROJECT GUTENBERG EBOOK PAULINE ET PASCAL BRUNO ***
- 33. CHEZ LES MÊMES EDITEURS: BIBLIOTHÈQUE LITTÉRAIRE ŒUVRES COMPLÈTES D’ALEXANDRE DUMAS Format in-18 anglais, à 2 francs le volume. EN VENTE: Le Comte de Monte-Cristo. 6 vol. Le Capitaine Paul. 1 — Le Chevalier d’Harmental. 2 — Les Trois Mousquetaires. 2 — Vingt Ans après. 3 — La Reine Margot. 2 — La Dame de Monsoreau. 3 — Quinze jours au Sinaï. 1 — Jacques Ortis. 1 — Le Chevalier de Maison-Rouge. 1 — Souvenirs d’Antony. 1 — Pauline et Pascal Bruno. 1 — Une fille du Régent. 1 — Ascanio. 2 — Sylvandire. 1 — Georges. 1 — Cécile. 1 — Isabel de Bavière. 2 —
- 34. Fernande. 1 — Amaury. 1 — SOUS PRESSE: Le Maître d’Armes. 1 vol. Paris.—Imp. Lacrampe fils et Comp., rue Damiette, 2.
- 35. PAULINE MURAT et PASCAL BRUNO PAR ALEXANDRE DUMAS PARIS MICHEL LÉVY FRÈRES, LIBRAIRES-ÉDITEURS
- 36. des Œuvres complètes d’Alexandre Dumas, DE LA BIBLIOTHÈQUE DRAMATIQUE ET DU THÉATRE DE VICTOR HUGO, Rue Vivienne, 1. 1848
- 38. PA U L IN E . I . Vers la fin de l’année 1834, nous étions réunis un samedi soir dans un petit salon attenant à la salle d’armes de Grisier, écoutant, le fleuret à la main et le cigare à la bouche, les savantes théories de notre professeur, interrompues de temps en temps par des anecdotes à l’appui, lorsque la porte s’ouvrit, et que Alfred de Nerval entra. Ceux qui ont lu mon Voyage en Suisse se rappelleront peut-être ce jeune homme qui servait de cavalier à une femme mystérieuse et voilée qui m’était apparue pour la première fois à Fluélen, lorsque je courais avec Francesco pour rejoindre la barque qui devait nous conduire à la pierre de Guillaume Tell: ils n’auront point oublié alors que, loin de m’attendre, Alfred de Nerval, que j’espérais avoir pour compagnon de voyage, avait hâté le départ des bateliers, et, quittant la rive au moment où j’en étais encore éloigné de trois cents pas, m’avait fait de la main un signe, à la fois d’adieu et d’amitié, que je traduisis par ces mots: «Pardon, cher ami, j’aurais grand plaisir à te voir, mais je ne suis pas seul, et...» A ceci j’avais répondu par un autre signe qui voulait dire: «Je comprends parfaitement.» Et je m’étais arrêté et incliné en marque d’obéissance à cette décision, si sévère qu’elle me parût; de sorte que, faute de barque et de bateliers, ce ne fut que le lendemain que je pus partir; de retour à l’hôtel, j’avais alors demandé si l’on connaissait cette femme, et l’on m’avait répondu que tout ce qu’on savait d’elle, c’est qu’elle paraissait fort souffrante, et qu’elle s’appelait Pauline.
- 39. J’avais oublié complétement cette rencontre, lorsqu’en allant visiter la source d’eau chaude qui alimente les bains de Pfeffers, je vis venir, peut-être se le rappellera-t-on encore, sous la longue galerie souterraine, Alfred de Nerval, donnant le bras à cette même femme que j’avais déjà entrevue à Fluélen, et qui, là, m’avait manifesté son désir de rester inconnue de la manière que j’ai racontée. Cette fois encore elle me parut désirer garder le même incognito, car son premier mouvement fut de retourner en arrière. Malheureusement le chemin sur lequel nous marchions ne permettait de s’écarter ni à droite ni à gauche: c’était une espèce de pont composé de deux planches humides et glissantes, qui, au lieu d’être jetées en travers d’un précipice, au fond duquel grondait la Tamina sur un lit de marbre noir, longeaient une des parois du souterrain, à quarante pieds à peu près au-dessus du torrent, soutenues par des poutres enfoncées dans le rocher. La mystérieuse compagne de mon ami pensa donc que toute fuite était impossible; alors, prenant son parti, elle baissa son voile, et continua de s’avancer vers moi. Je racontai alors la singulière impression que me fit cette femme blanche et légère comme une ombre, marchant au bord de l’abîme sans plus paraître s’en inquiéter que si elle appartenait déjà à un autre monde. En la voyant s’approcher, je me rangeai contre la muraille, afin d’occuper le moins de place possible. Alfred voulut la faire passer seule; mais elle refusa de quitter son bras, de sorte que nous nous trouvâmes un instant à trois sur une largeur de deux pieds tout au plus; mais cet instant fut prompt comme un éclair: cette femme étrange, pareille à une de ces fées qui se penchent au bord des torrens et font flotter leur écharpe dans l’écume des cascades, s’inclina sur le précipice, et passa comme par miracle, mais pas si rapidement encore que je ne pusse entrevoir son visage calme et doux, quoique pâle et amaigri par la souffrance. Alors il me sembla que ce n’était point la première fois que je voyais cette figure; il s’éveilla dans mon esprit un souvenir vague d’une autre époque, une réminiscence de salons, de bals, de fêtes; il me semblait que j’avais connu cette femme, au visage si défait et si triste aujourd’hui, joyeuse, rougissante et couronnée de fleurs, emportée au milieu des parfums et de la musique dans quelque valse langoureuse ou quelque galop bondissant: où cela? je
- 40. n’en savais plus rien; à quelle époque? il m’était impossible de le dire; c’était une vision, un rêve, un écho de ma mémoire, qui n’avait rien de précis et de réel, et qui m’échappait comme si j’eusse voulu saisir une vapeur. Je revins en me promettant de la revoir, dussé-je être indiscret pour parvenir à ce but; mais, à mon retour, quoique je n’eusse été absent qu’une demi-heure, ni Alfred ni elle n’étaient déjà plus aux bains de Pfeffers. Deux mois s’étaient écoulés depuis cette seconde rencontre; je me trouvais à Baveno, près du lac Majeur: c’était par une belle soirée d’automne; le soleil venait de disparaître derrière la chaîne des Alpes, et l’ombre montait à l’orient, qui commençait à se parsemer d’étoiles. La fenêtre de ma chambre donnait de plain-pied sur une terrasse toute couverte de fleurs; j’y descendis, et je me trouvai au milieu d’une forêt de lauriers-roses, de myrtes et d’orangers. C’est une si douce chose que les fleurs, que ce n’est point encore assez d’en être entouré, on veut en jouir de plus près, et, quelque part qu’on en trouve, fleurs des champs, fleurs des jardins, l’instinct de l’enfant, de la femme et de l’homme, est de les arracher à leur tige, et d’en faire un bouquet dont le parfum les suive, et dont l’éclat soit à eux. Aussi ne résistai-je pas à la tentation; je brisai quelques branches embaumées, et j’allai m’appuyer sur la balustrade de granit rose qui domine le lac, dont elle n’est séparée que par la grande route qui va de Genève à Milan. J’y fus à peine, que la lune se leva du côté de Sesto, et que ses rayons commencèrent à glisser aux flancs des montagnes qui bornaient l’horizon et sur l’eau qui dormait à mes pieds, resplendissante et tranquille comme un immense miroir: tout était calme; aucun bruit ne venait de la terre, du lac, ni du ciel, et la nuit commençait sa course dans une majestueuse et mélancolique sérénité. Bientôt, d’un massif d’arbres qui s’élevait à ma gauche, et dont les racines baignaient dans l’eau, le chant d’un rossignol s’élança harmonieux et tendre; c’était le seul son qui veillât; il se soutint un instant, brillant et cadencé, puis tout-à-coup il s’arrêta à la fin d’une roulade. Alors, comme si ce bruit en eût éveillé un autre d’une nature bien différente, le roulement lointain d’une voiture se fit entendre venant de Doma d’Ossola, puis le chant du rossignol reprit, et je n’écoutai
- 41. plus que l’oiseau de Juliette. Lorsqu’il cessa, j’entendis de nouveau la voiture plus rapprochée; elle venait rapidement; cependant, si rapide que fût sa course, mon mélodieux voisin eut encore le temps de reprendre sa nocturne prière. Mais cette fois, à peine eut-il lancé sa dernière note, qu’au tournant de la route j’aperçus une chaise de poste qui roulait, emportée par le galop de deux chevaux, sur le chemin qui passait devant l’auberge. A deux cents pas de nous, le postillon fit claquer bruyamment son fouet, afin d’avertir son confrère de son arrivée. En effet, presque aussitôt la grosse porte de l’auberge grinça sur ses gonds, et un nouvel attelage en sortit; au même instant, la voiture s’arrêta au-dessous de la terrasse à la balustrade de laquelle j’étais accoudé. La nuit, comme je l’ai dit, était si pure, si transparente et si parfumée, que les voyageurs, pour jouir des douces émanations de l’air, avaient abaissé la capote de la calèche. Ils étaient deux, un jeune homme et une jeune femme: la jeune femme enveloppée dans un grand châle ou dans un manteau, et la tête renversée en arrière sur le bras du jeune homme qui la soutenait. En ce moment le postillon sortit avec une lumière pour allumer les lanternes de la voiture, un rayon de clarté passa sur la figure des voyageurs, et je reconnus Alfred de Nerval et Pauline. Toujours lui et toujours elle! il semblait qu’une puissance plus intelligente que le hasard nous poussait à la rencontre les uns des autres. Toujours elle, mais si changée encore depuis Pfeffers, si pâle, si mourante, que ce n’était plus qu’une ombre; et cependant ces traits flétris rappelèrent encore à mon esprit cette vague image de femme qui dormait au fond de ma mémoire, et qui, à chacune de ces apparitions, montait à sa surface et glissait sur ma pensée comme sur le brouillard une rêverie d’Ossian. J’étais tout près d’appeler Alfred, mais je me rappelai combien sa compagne désirait ne pas être vue. Et pourtant un sentiment de si mélancolique pitié m’entraînait vers elle, que je voulus qu’elle sût du moins que quelqu’un priait pour que son âme tremblante et prête à s’envoler n’abandonnât pas sitôt avant l’heure le corps gracieux qu’elle animait. Je pris une carte de visite dans ma poche; j’écrivis au dos avec mon crayon: «Dieu garde les voyageurs, console les affligés et
- 42. guérisse les souffrans.» Je mis la carte au milieu des branches d’orangers, de myrtes et de roses que j’avais cueillies, et je laissai tomber le bouquet dans la voiture. Au même instant le postillon repartit, mais pas si rapidement que je n’aie eu le temps de voir Alfred se pencher en dehors de la voiture afin d’approcher ma carte de la lumière. Alors il se retourna de mon côté, me fit un signe de la main, et la calèche disparut à l’angle de la route. Le bruit de la voiture s’éloigna, mais sans être interrompu cette fois par le chant du rossignol. J’eus beau me tourner du côté du buisson et rester une heure encore sur la terrasse, j’attendis vainement. Alors une pensée profondément triste me prit: je me figurai que cet oiseau qui avait chanté, c’était l’âme de la jeune fille qui dit son cantique d’adieu à la terre, et que, puisqu’il ne chantait plus, c’est qu’elle était déjà remontée au ciel. La situation ravissante de l’auberge, placée entre les Alpes qui finissent et l’Italie qui commence, ce spectacle calme et en même temps animé du lac Majeur, avec ses trois îles, dont l’une est un jardin, l’autre un village et la troisième un palais; ces premières neiges de l’hiver qui couvraient les montagnes, et ces dernières chaleurs de l’automne qui venaient de la Méditerranée, tout cela me retint huit jours à Baveno; puis je partis pour Arona, et d’Arona pour Sesto Calende. Là m’attendait un dernier souvenir de Pauline; là, l’étoile que j’avais vue filer à travers le ciel s’était éteinte; là, ce pied si léger au bord du précipice avait heurté la tombe; et jeunesse usée, beauté flétrie, cœur brisé, tout s’était englouti sous une pierre, voile du sépulcre, qui, fermé aussi mystérieusement sur ce cadavre que le voile de la vie avait été tiré sur le visage, n’avait laissé pour tout renseignement à la curiosité du monde que le prénom de Pauline. J’allai voir cette tombe: au contraire des tombes italiennes, qui sont dans les églises, celle-ci s’élevait dans un charmant jardin, au haut d’une colline boisée, sur le versant qui regardait et dominait le lac. C’était le soir; la pierre commençait à blanchir aux rayons de la lune; je m’assis près d’elle, forçant ma pensée à ressaisir tout ce
- 43. qu’elle avait de souvenirs épars et flottans de cette jeune femme; mais cette fois encore ma mémoire fut rebelle; je ne pus réunir que des vapeurs sans forme, et non une statue aux contours arrêtés, et je renonçai à pénétrer ce mystère jusqu’au jour où je retrouverais Alfred de Nerval. On comprendra facilement maintenant combien son apparition inattendue, au moment où je songeais le moins à lui, vint frapper tout à la fois mon esprit, mon cœur et mon imagination d’idées nouvelles; en un instant je revis tout: cette barque qui m’échappait sur le lac; ce pont souterrain, pareil à un vestibule de l’enfer, où les voyageurs semblent des ombres; cette petite auberge de Baveno, au pied de laquelle était passée la voiture mortuaire; puis enfin cette pierre blanchissante, où, aux rayons de la lune glissant entre les branches des orangers et des lauriers-roses, on peut lire, pour toute épitaphe, le prénom de cette femme morte si jeune et probablement si malheureuse. Aussi m’élançai-je vers Alfred comme un homme enfermé depuis longtemps dans un souterrain s’élance à la lumière qui entre par une porte que l’on ouvre; il sourit tristement en me tendant la main, comme pour me dire qu’il me comprenait; et ce fut alors moi qui fis un mouvement en arrière et qui me repliai en quelque sorte sur moi- même, afin que Alfred, vieil ami de quinze ans, ne prît pas pour un simple mouvement de curiosité le sentiment qui m’avait poussé au- devant de lui. Il entra. C’était un des bons élèves de Grisier, et cependant depuis près de trois ans il n’avait point paru à la salle d’armes. La dernière fois qu’il y était venu, il avait un duel pour le lendemain, et, ne sachant encore à quelle arme il se battrait, il venait, à tout hasard, se refaire la main avec le maître. Depuis ce temps, Grisier ne l’avait pas revu; il avait entendu dire seulement qu’il avait quitté la France et habitait Londres. Grisier, qui tient à la réputation de ses élèves autant qu’à la sienne, n’eut pas plutôt échangé avec lui les complimens d’usage, qu’il lui mit un fleuret dans la main, lui choisit parmi nous un
- 44. adversaire de sa force; c’était, je m’en souviens, ce pauvre Labattut, qui partait pour l’Italie, et qui, lui aussi, allait trouver à Pise une tombe ignorée et solitaire. A la troisième passe, le fleuret de Labattut rencontra la poignée de l’arme de son adversaire, et, se brisant à deux pouces au- dessous du bouton, alla en passant à travers la garde, déchirer la manche de sa chemise, qui se teignit de sang. Labattut jeta aussitôt son fleuret; il croyait, comme nous, Alfred sérieusement blessé. Heureusement ce n’était qu’une égratignure; mais, en relevant la manche de sa chemise, Alfred nous découvrit une autre cicatrice qui avait dû être plus sérieuse; une balle de pistolet lui avait traversé les chairs de l’épaule. —Tiens! lui dit Grisier avec étonnement, je ne vous savais pas cette blessure? C’est que Grisier nous connaissait tous, comme une nourrice son enfant; pas un de ses élèves n’avait une piqûre sur le corps dont il ne sût la date et la cause. Il écrirait une histoire amoureuse bien amusante et bien scandaleuse, j’en suis sûr, s’il voulait raconter celle des coups d’épée dont il sait les antécédens; mais cela ferait trop de bruit dans les alcôves, et, par contre-coup, trop de tort à son établissement; il en fera des mémoires posthumes. —C’est, lui répondit Alfred, que je l’ai reçue le lendemain du jour où je suis venu faire assaut avec vous, et que, le jour où je l’ai reçue, je suis parti pour l’Angleterre. —Je vous avais bien dit de ne pas vous battre au pistolet. Thèse générale: l’épée est l’arme du brave et du gentilhomme, l’épée est la relique la plus précieuse que l’histoire conserve des grands hommes qui ont illustré la patrie: on dit l’épée de Charlemagne, l’épée de Bayard, l’épée de Napoléon, qui est-ce qui a jamais parlé de leur pistolet? Le pistolet est l’arme du brigand; c’est le pistolet sous la gorge qu’on fait signer de fausses lettres de change; c’est le pistolet à la main qu’on arrête une diligence au coin d’un bois; c’est avec un pistolet que le banqueroutier se brûle la cervelle... Le pistolet!... fi
- 45. donc!... L’épée, à la bonne heure! c’est la compagne, c’est la confidente, c’est l’amie de l’homme; elle garde son honneur ou elle le venge. —Eh bien! mais, avec cette conviction, répondit en souriant Alfred, comment vous êtes-vous battu il y a deux ans au pistolet? —Moi, c’est autre chose: je dois me battre à tout ce qu’on veut; je suis maître d’armes; et puis il y a des circonstances où l’on ne peut pas refuser les conditions qu’on vous impose... —Eh bien! je me suis trouvé dans une de ces circonstances, mon cher Grisier, et vous voyez que je ne m’en suis pas mal tiré... —Oui, avec une balle dans l’épaule. —Cela valait toujours mieux qu’une balle dans le cœur. —Et peut-on savoir la cause de ce duel? —Pardonnez-moi, mon cher Grisier, mais toute cette histoire est encore un secret; plus tard, vous la connaîtrez. —Pauline?... lui dis-je tout bas. —Oui, me répondit-il. —Nous la connaîtrons, bien sûr?... dit Grisier. —Bien sûr, reprit Alfred; et la preuve, c’est que j’emmène souper Alexandre, et que je la lui raconterai ce soir; de sorte qu’un beau jour, lorsqu’il n’y aura plus d’inconvénient à ce qu’elle paraisse, vous la trouverez dans quelque volume intitulé: Contes bruns ou Contes bleus. Prenez donc patience jusque-là. Force fut donc à Grisier de se résigner. Alfred m’emmena souper comme il me l’avait offert, et me raconta l’histoire de Pauline. Aujourd’hui le seul inconvénient qui existât à sa publication a disparu. La mère de Pauline est morte, et avec elle s’est éteinte la famille et le nom de cette malheureuse enfant, dont les aventures
- 46. semblent empruntées à une époque ou à une localité bien étrangères à celles où nous vivons.
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