\contentsline {chapter}{\numberline {1}Discrete Probability Distributions}{1}{chapter.1}
\contentsline {section}{\numberline {1.1}Simulation of Discrete Probabilities}{1}{section.1.1}
\contentsline {section}{\numberline {1.2}Discrete Probability Distributions}{18}{section.1.2}
\contentsline {chapter}{\numberline {2}Continuous Probability Densities}{41}{chapter.2}
\contentsline {section}{\numberline {2.1}Simulation of Continuous Probabilities}{41}{section.2.1}
\contentsline {section}{\numberline {2.2}Continuous Density Functions}{55}{section.2.2}
\contentsline {chapter}{\numberline {3}Combinatorics}{77}{chapter.3}
\contentsline {section}{\numberline {3.1}Permutations}{77}{section.3.1}
\contentsline {section}{\numberline {3.2}Combinations}{95}{section.3.2}
\contentsline {section}{\numberline {3.3}Card Shuffling}{122}{section.3.3}
\contentsline {chapter}{\numberline {4}Conditional Probability}{137}{chapter.4}
\contentsline {section}{\numberline {4.1}Discrete Conditional Probability}{137}{section.4.1}
\contentsline {section}{\numberline {4.2}Continuous Conditional Probability}{167}{section.4.2}
\contentsline {section}{\numberline {4.3}Paradoxes}{181}{section.4.3}
\contentsline {chapter}{\numberline {5}Distributions and Densities}{189}{chapter.5}
\contentsline {section}{\numberline {5.1}Important Distributions}{189}{section.5.1}
\contentsline {section}{\numberline {5.2}Important Densities}{211}{section.5.2}
\contentsline {chapter}{\numberline {6}Expected Value and Variance}{231}{chapter.6}
\contentsline {section}{\numberline {6.1}Expected Value}{231}{section.6.1}
\contentsline {section}{\numberline {6.2}Variance of Discrete Random Variables}{263}{section.6.2}
\contentsline {section}{\numberline {6.3}Continuous Random Variables}{275}{section.6.3}
\contentsline {chapter}{\numberline {7}Sums of Random Variables}{293}{chapter.7}
\contentsline {section}{\numberline {7.1}Sums of Discrete Random Variables}{293}{section.7.1}
\contentsline {section}{\numberline {7.2}Sums of Continuous Random Variables}{299}{section.7.2}
\contentsline {chapter}{\numberline {8}Law of Large Numbers}{313}{chapter.8}
\contentsline {section}{\numberline {8.1}Discrete Random Variables}{313}{section.8.1}
\contentsline {section}{\numberline {8.2}Continuous Random Variables}{324}{section.8.2}
\contentsline {chapter}{\numberline {9}Central Limit Theorem}{333}{chapter.9}
\contentsline {section}{\numberline {9.1}Bernoulli Trials}{333}{section.9.1}
\contentsline {section}{\numberline {9.2}Discrete Independent Trials}{348}{section.9.2}
\contentsline {section}{\numberline {9.3}Continuous Independent Trials}{364}{section.9.3}
\contentsline {chapter}{\numberline {10}Generating Functions}{373}{chapter.10}
\contentsline {section}{\numberline {10.1}Discrete Distributions}{373}{section.10.1}
\contentsline {section}{\numberline {10.2}Branching Processes}{385}{section.10.2}
\contentsline {section}{\numberline {10.3}Continuous Densities}{402}{section.10.3}
\contentsline {chapter}{\numberline {11}Markov Chains}{413}{chapter.11}
\contentsline {section}{\numberline {11.1}Introduction}{413}{section.11.1}
\contentsline {section}{\numberline {11.2}Absorbing Markov Chains}{424}{section.11.2}
\contentsline {section}{\numberline {11.3}Ergodic Markov Chains}{441}{section.11.3}
\contentsline {section}{\numberline {11.4}Fundamental Limit Theorem}{456}{section.11.4}
\contentsline {section}{\numberline {11.5}Mean First Passage Time}{461}{section.11.5}
\contentsline {chapter}{\numberline {12}Random Walks}{481}{chapter.12}
\contentsline {section}{\numberline {12.1}Random Walks in Euclidean Space}{481}{section.12.1}
\contentsline {section}{\numberline {12.2}Gambler's Ruin}{496}{section.12.2}
\contentsline {section}{\numberline {12.3}Arc Sine Laws}{503}{section.12.3}