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CHAPTER 1 The Random Process and Gambling Theory 3
Independent versus Dependent Trials Processes 5
Mathematical Expectation 6
Exact Sequences, Possible Outcomes, and the
Normal Distribution 8
Possible Outcomes and Standard Deviations 11
The House Advantage 15
Mathematical Expectation Less than Zero Spells Disaster 18
Baccarat 19
Numbers 20
Pari-Mutuel Betting 21
Winning and Losing Streaks in the Random Process 24
Determining Dependency 25
The Runs Test, Z Scores, and Confidence Limits 27
The Linear Correlation Coefficient 32
CHAPTER 2 Probability Distributions 43
The Basics of Probability Distributions 43
Descriptive Measures of Distributions 45
Moments of a Distribution 47
The Normal Distribution 52
vii
viii THE HANDBOOK OF PORTFOLIO MATHEMATICS
The Central Limit Theorem 52
Working with the Normal Distribution 54
Normal Probabilities 59
Further Derivatives of the Normal 65
The Lognormal Distribution 67
The Uniform Distribution 69
The Bernoulli Distribution 71
The Binomial Distribution 72
The Geometric Distribution 78
The Hypergeometric Distribution 80
The Poisson Distribution 81
The Exponential Distribution 85
The Chi-Square Distribution 87
The Chi-Square “Test” 88
The Student’s Distribution 92
The Multinomial Distribution 95
The Stable Paretian Distribution 96
CHAPTER 3 Reinvestment of Returns and Geometric
Growth Concepts 99
To Reinvest Trading Profits or Not 99
Measuring a Good System for Reinvestment—The
Geometric Mean 103
Estimating the Geometric Mean 107
How Best to Reinvest 109
CHAPTER 4 Optimal f 117
Optimal Fixed Fraction 117
Asymmetrical Leverage 118
Kelly 120
Finding the Optimal f by the Geometric Mean 122
To Summarize Thus Far 125
How to Figure the Geometric Mean Using
Spreadsheet Logic 127
Geometric Average Trade
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