Elementary Probability Theory

PREFACE TO THE FOURTH EDITION
PROLOGUE TO INTRODUCTION TO MATHEMATICAL FINANCE
1 SET
1.1 Sample sets
1.2 Operations with sets
1.3 Various relations
1.4 Indicator
Exercises
2 PROBABILITY
2.1 Examples of probability
2.2 Definition and illustrations
2.3 Deductions from the axioms
2.4 Independent events
2.5 Arithmetical density
Exercises
3 COUNTING
3.1 Fundamental rule
3.2 Diverse ways of sampling
3.3 Allocation models; binomial coefficients
3.4 How to solve it
Exercises
4 RANDOM VARIABLES
4.1 What is a random variable?
4.2 How do random variables come about?
4.3 Distribution and expectation
4.4 Integer-valued random variables
4.5 Random variables with densities
4.6 General case
Exercises
APPENDIX 1: BOREL FIELDS AND GENERAL RANDOM VARIABLES
5 CONDITIONING AND INDEPENDENCE
5.1 Examples of conditioning
5.2 Basic formulas
5.3 Sequential sampling
5.4 Pólya's urn scheme
5.5 Independence and relevance
5.6 Genetical models
Exercises
6 MEAN, VARIANCE, AND TRANSFORMS
6.1 Basic properties of expectation
6.2 The density case
6.3 Multiplication theorem; variance and covariance
6.4 Multinomial distribution
6.5 Generating function and the like
Exercises
7 POISSON AND NORMAL DISTRIBUTIONS
7.1 Models for Poisson distribution
7.2 Poisson process
7.3 From binomial to normal
7.4 Normal distribution
7.5 Central limit theorem
7.6 Law of large numbers
Exercises
APPENDIX 2: STIRLING'S FORMULA AND DE MOIVRE-LAPLACE'S THEOREM
8 FROM RANDOM WALKS TO MARKOV CHAINS
8.1 Problems of the wanderer or gambler
8.2 Limiting schemes
8.3 Transition probabilities
8.4 Basic structure of Markov chains
8.5 Further developments
8.6 Steady state
8.7 Winding up (or down?)
Exercises
APPENDIX 3: MARTINGALE
9 MEAN-VARIANCE PRICING MODEL
9.1 An investments primer
9.2 Asset return and risk
9.3 Portfolio allocation
9.4 Diversification
9.5 Mean-variance optimization
9.6 Asset return distributions
9.7 Stable probability distributions
Exercises
APPENDIX 4: PARETO AND STABLE LAWS
10 OPTION PRICING THEORY
10.1 Options basics
10.2 Arbitrage-free pricing: 1-period model
10.3 Arbitrage-free pricing: N-period model
10.4 Fundamental asset pricing theorems
Exercises
GENERAL REFERENCES
ANSWERS TO PROBLEMS
VALUES OF THE STANDARD NORMAL DISTRIBUTION FUNCTION
INDEX