Upper level undergraduate probability with actuarial and financial applications Richard F. Bass ...

181 Pages·2017·1.32 MB·English

Upper level undergraduate probability with
actuarial and financial applications
Richard F. Bass
Patricia Alonso-Ruiz, Fabrice Baudoin, Maria Gordina,
Phanuel Mariano, Oleksii Mostovyi, Ambar Sengupta,
Alexander Teplyaev, Emiliano Valdez
© Copyright 2013 Richard F. Bass
© Copyright 2017 University of Connecticut, Department of Mathematics
© Copyright 2018 University of Connecticut, Department of Mathematics
Acknowledgments. The authors are very grateful to Joe Chen, Tom Laetsch, Tom
Roby, Linda Westrick for their important contributions to this project, and to all the Uni-
versity of Connecticut students in Math 3160 for their tireless work in the course.
We gratefully acknowledge the generous support from the University of Connecticut Open
and Affordable Initiative, the Davis Educational Foundation, the UConn Co-op Bookstore,
and the University of Connecticut Libraries.
Preface
This textbook has been created as a part of the University of Connecticut Open and Afford-
able Initiative, which in turn was a response to the Connecticut State Legislature Special Act
No. 15-18 (House Bill 6117), An Act Concerning the Use of Digital Open-Source Textbooks
in Higher Education. At the University of Connecticut this initiative was strongly supported
by the UConn Bookstore and the University of Connecticut Libraries. Generous external
support was provided by the Davis Educational Foundation.
Even before this initiative, our department had a number of freely available and internal
resources for Math 3160, our basic probability course. This included lecture notes prepared
by Richard Bass, the Board of Trustees Distinguished Professor of Mathematics. Therefore,
it was natural to extend the lecture notes into a complete textbook for the course. Two
aspects of the courses were taken into account. On the one hand, the course is taken by
many students who are interested in the financial and actuarial careers. On the other hand,
this course has multivariable calculus as a prerequisite, which is not common for most of
the undergraduate probability courses taught at other universities. The 2018 edition of the
textbook has 4 parts divided into 15 chapters. The first 3 parts consist of required material
for Math 3160, and the 4th part contains optional material for this course.
Our textbook has been used in classrooms during 3 semesters at UConn, and received over-
whelmingly positive feedback from students. However, we are still working on improving the
text, and will be grateful for comments and suggestions.
May 2018.
iii

Contents
Acknowledgments ii
Preface iii
Part 1. Discrete Random Variables 1
Chapter 1. Combinatorics 3
1.1. Introduction 3
1.2. Further examples and explanations 6
1.2.1. Counting Principle revisited 6
1.2.2. Permutations 7
1.2.3. Combinations 8
1.2.4. Multinomial Coefficients 10
1.3. Exercises 11
1.4. Selected solutions 13
Chapter 2. The probability set-up 15
2.1. Introduction and basic theory 15
2.2. Further examples and applications 18
2.2.1. Events 18
2.2.2. Axioms of probability and their consequences 18
2.2.3. Uniform discrete distribution 20
2.3. Exercises 21
2.4. Selected solutions 23
Chapter 3. Independence 27
3.1. Introduction 27
3.2. Further examples and explanations 30
3.2.1. Independent Events 30
3.2.2. Bernoulli trials 31
3.3. Exercises 32
3.4. Selected solutions 34
Chapter 4. Conditional probability 37
4.1. Introduction 37
4.2. Further examples and applications 40
4.2.1. Conditional Probabilities 40
4.2.2. Bayes’ rule 42
4.3. Exercises 44
4.4. Selected solutions 46
v
vi CONTENTS
Chapter 5. Random variables 49
5.1. Introduction 49
5.2. Further examples and applications 53
5.2.1. Random Variables 53
5.2.2. Discrete Random Variables 53
5.2.3. Expected Value 54
5.2.4. The cumulative distribution function (c.d.f.) 55
5.2.5. Expectated Value of Sums of Random Variables 57
5.2.6. Expectation of a Function of a Random Variable 59
5.2.7. Variance 60
5.3. Exercises 61
5.4. Selected solutions 63
Chapter 6. Some discrete distributions 67
6.1. Introduction 67
6.2. Further examples and applications 70
6.2.1. Bernouli and Binomial Random Variables 70
6.2.2. The Poisson Distribution 70
6.2.3. Table of distributions 72
6.3. Exercises 73
6.4. Selected solutions 75
Part 2. Continuous Random Variables 79
Chapter 7. Continuous distributions 81
7.1. Introduction 81
7.2. Further examples and applications 84
7.3. Exercises 86
7.4. Selected solutions 88
Chapter 8. Normal distribution 91
8.1. Introduction 91
8.2. Further examples and applications 94
8.3. Exercises 95
8.4. Selected solutions 96
Chapter 9. Normal approximation to the binomial 99
9.1. Introduction 99
9.2. Exercises 101
9.3. Selected solutions 102
Chapter 10. Some continuous distributions 103
10.1. Introduction 103
10.2. Further examples and applications 105
10.3. Exercises 106
10.4. Selected solutions 107
Part 3. Multivariate Discrete and Continuous Random Variables 109
CONTENTS vii
Chapter 11. Multivariate distributions 111
11.1. Introduction 111
11.2. Further examples and applications 117
11.3. Exercises 121
11.4. Selected solutions 123
Chapter 12. Expectations 129
12.1. Introduction 129
12.2. Further examples and applications 131
12.2.1. Expectation and variance 131
12.2.2. Correlation 132
12.2.3. Conditional expectation 134
12.3. Exercises 136
12.4. Selected solutions 137
Chapter 13. Moment generating functions 139
13.1. Introduction 139
13.2. Further examples and applications 141
13.3. Exercises 143
13.4. Selected solutions 144
Chapter 14. Limit laws 147
14.1. Introduction 147
14.2. Further examples and applications 150
14.3. Exercises 151
14.4. Selected solutions 152
Part 4. Applications of Probability Distributions and Random Variables 155
Chapter 15. Application of Probability in Finance 157
15.1. Introduction 157
15.1.1. The simple coin toss game 157
15.1.2. The coin toss game stopped at zero 158
15.1.3. The coin toss game with borrowing at zero 159
15.1.4. Probability to win $N before reaching as low as $L 160
15.1.5. Expected playing time 161
15.1.6. Doubling the money coin toss game 162
15.2. Exercises on simple coin toss games 163
15.3. Problems motivated by the American options pricing 164
15.4. Problems about the Black-Scholes-Merton pricing formula 167
15.5. Selected solutions 168
Appendix: a table of common distributions and a table of the standard normal
cumulative distribution function 171

Part 1
Discrete Random Variables