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Introduction to Probability
Charles M. Grinstead
Swarthmore College
J. Laurie Snell
Dartmouth College
To our wives
and in memory of
Reese T. Prosser
Contents
1
Discrete Probability Distributions
1
1.1
Simulation of Discrete Probabilities ...................
1
1.2
Discrete Probability Distributions .................... 18
2
Continuous Probability Densities
41
2.1
Simulation of Continuous Probabilities ................. 41
2.2
Continuous Density Functions ...................... 55
3
Combinatorics
75
3.1
Permutations
............................... 75
3.2
Combinations ............................... 92
3.3
Card Shu²ing ............................... 120
4
Conditional Probability
133
4.1
Discrete Conditional Probability
.................... 133
4.2
Continuous Conditional Probability ................... 162
4.3
Paradoxes ................................. 175
5
Distributions and Densities
183
5.1
Important Distributions
......................... 183
5.2
Important Densities
........................... 205
6
Expected Value and Variance
225
6.1
Expected Value .............................. 225
6.2
Variance of Discrete Random Variables ................. 257
6.3
Continuous Random Variables ...................... 268
7
Sums of Random Variables
285
7.1
Sums of Discrete Random Variables
.................. 285
7.2
Sums of Continuous Random Variables ................. 291
8
Law of Large Numbers
305
8.1
Discrete Random Variables
....................... 305
8.2
Continuous Random Variables ...................... 316
v
vi
CONTENTS
9
Central Limit Theorem
325
9.1
Bernoulli Trials .............................. 325
9.2
Discrete Independent Trials
....................... 340
9.3
Continuous Independent Trials
..................... 355
10 Generating Functions 365
10.1 Discrete Distributions .......................... 365
10.2 Branching Processes ........................... 377
10.3 Continuous Densities ........................... 394
11 Markov Chains 405
11.1 Introduction ................................ 405
11.2 Absorbing Markov Chains ........................ 415
11.3 Ergodic Markov Chains ......................... 433
11.4 Fundamental Limit Theorem
...................... 447
11.5 Mean First Passage Time
........................ 452
12 Random Walks 471
12.1 Random Walks in Euclidean Space ................... 471
12.2 Gambler's Ruin .............................. 486
12.3 Arc Sine Laws ............................... 493
Appendices
499
A
Normal Distribution Table ........................ 499
B
Galton's Data ............................... 500
C
Life Table ................................. 501
Index
503
Preface
Probability theory began in seventeenth century France when the two great French
mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two prob-
lems from games of chance. Problems like those Pascal and Fermat solved continued
to in°uence such early researchers as Huygens, Bernoulli, and DeMoivre in estab-
lishing a mathematical theory of probability. Today, probability theory is a well-
established branch of mathematics that ¯nds applications in every area of scholarly
activity from music to physics, and in daily experience from weather prediction to
predicting the risks of new medical treatments.
This text is designed for an introductory probability course taken by sophomores,
juniors, and seniors in mathematics, the physical and social sciences, engineering,
and computer science. It presents a thorough treatment of probability ideas and
techniques necessary for a ¯rm understanding of the subject. The text can be used
in a variety of course lengths, levels, and areas of emphasis.
For use in a standard one-term course, in which both discrete and continuous
probability is covered, students should have taken as a prerequisite two terms of
calculus, including an introduction to multiple integrals. In order to cover Chap-
ter 11, which contains material on Markov chains, some knowledge of matrix theory
is necessary.
The text can also be used in a discrete probability course. The material has been
organized in such a way that the discrete and continuous probability discussions are
presented in a separate, but parallel, manner. This organization dispels an overly
rigorous or formal view of probability and o®ers some strong pedagogical value
in that the discrete discussions can sometimes serve to motivate the more abstract
continuous probability discussions. For use in a discrete probability course, students
should have taken one term of calculus as a prerequisite.
Very little computing background is assumed or necessary in order to obtain full
bene¯ts from the use of the computing material and examples in the text. All of
the programs that are used in the text have been written in each of the languages
TrueBASIC, Maple, and Mathematica.
This book is on the Web at http://www.dartmouth.edu/~chance, and is part of
the Chance project, which is devoted to providing materials for beginning courses in
probability and statistics. The computer programs, solutions to the odd-numbered
exercises, and current errata are also available at this site. Instructors may obtain
all of the solutions by writing to either of the authors, at jlsnell@dartmouth.edu and
cgrinst1@swarthmore.edu. It is our intention to place items related to this book at
vii
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