\contentsline {chapter}{Preface}{vii}
\contentsline {chapter}{\numberline {1}Discrete Probability Distributions}{1}
\contentsline {section}{\numberline {1.1}Simulation of Discrete Probabilities}{1}
\contentsline {section}{\numberline {1.2}Discrete Probability Distributions}{18}
\contentsline {chapter}{\numberline {2}Continuous Probability Densities}{41}
\contentsline {section}{\numberline {2.1}Simulation of Continuous Probabilities}{41}
\contentsline {section}{\numberline {2.2}Continuous Density Functions}{55}
\contentsline {chapter}{\numberline {3}Combinatorics}{75}
\contentsline {section}{\numberline {3.1}Permutations}{75}
\contentsline {section}{\numberline {3.2}Combinations}{92}
\contentsline {section}{\numberline {3.3}Card Shuffling}{120}
\contentsline {chapter}{\numberline {4}Conditional Probability}{133}
\contentsline {section}{\numberline {4.1}Discrete Conditional Probability}{133}
\contentsline {section}{\numberline {4.2}Continuous Conditional Probability}{162}
\contentsline {section}{\numberline {4.3}Paradoxes}{175}
\contentsline {chapter}{\numberline {5}Distributions and Densities}{183}
\contentsline {section}{\numberline {5.1}Important Distributions}{183}
\contentsline {section}{\numberline {5.2}Important Densities}{205}
\contentsline {chapter}{\numberline {6}Expected Value and Variance}{225}
\contentsline {section}{\numberline {6.1}Expected Value}{225}
\contentsline {section}{\numberline {6.2}Variance of Discrete Random Variables}{257}
\contentsline {section}{\numberline {6.3}Continuous Random Variables}{268}
\contentsline {chapter}{\numberline {7}Sums of Random Variables}{285}
\contentsline {section}{\numberline {7.1}Sums of Discrete Random Variables}{285}
\contentsline {section}{\numberline {7.2}Sums of Continuous Random Variables}{291}
\contentsline {chapter}{\numberline {8}Law of Large Numbers}{305}
\contentsline {section}{\numberline {8.1}Discrete Random Variables}{305}
\contentsline {section}{\numberline {8.2}Continuous Random Variables}{316}
\contentsline {chapter}{\numberline {9}Central Limit Theorem}{325}
\contentsline {section}{\numberline {9.1}Bernoulli Trials}{325}
\contentsline {section}{\numberline {9.2}Discrete Independent Trials}{340}
\contentsline {section}{\numberline {9.3}Continuous Independent Trials}{356}
\contentsline {chapter}{\numberline {10}Generating Functions}{365}
\contentsline {section}{\numberline {10.1}Discrete Distributions}{365}
\contentsline {section}{\numberline {10.2}Branching Processes}{376}
\contentsline {section}{\numberline {10.3}Continuous Densities}{393}
\contentsline {chapter}{\numberline {11}Markov Chains}{405}
\contentsline {section}{\numberline {11.1}Introduction}{405}
\contentsline {section}{\numberline {11.2}Absorbing Markov Chains}{416}
\contentsline {section}{\numberline {11.3}Ergodic Markov Chains}{433}
\contentsline {section}{\numberline {11.4}Fundamental Limit Theorem}{447}
\contentsline {section}{\numberline {11.5}Mean First Passage Time}{452}
\contentsline {chapter}{\numberline {12}Random Walks}{471}
\contentsline {section}{\numberline {12.1}Random Walks in Euclidean Space}{471}
\contentsline {section}{\numberline {12.2}Gambler's Ruin}{486}
\contentsline {section}{\numberline {12.3}Arc Sine Laws}{493}
\contentsline {chapter}{Appendices}{499}
\contentsline {chapter}{Index}{503}