Contents
Preface
Chapter 1. Combinatorics and a Few Calculus Facts
1.1 Combinatorial analysis
1.2 The exponential and logarithmic functions
Chapter 2. Fundamentals of Probability
2.1 Foundation of probability
2.2 The concept of conditional probability
2.3 The law of conditional probability
2.4 Bayesian approach to inference
2.4.1 Real-life cases of Bayesian thinking
2.4.2 Bayesian statistics vs. classical statistics
2.4.3 Naive Bayes in data analysis
2.5 The concept of random variable
2.6 Expected value and standard deviation
2.7 Independent random variables and the square root law
2.8 Generating functions
2.9 Inequalities and the law of large numbers
2.9.1 Kelly formula in gambling and investment
2.10 Additional material
2.10.1 Covariance and correlation
2.10.2 Conditional expectation
2.10.3 Logistic regression in data analysis
Chapter 3. Useful Probability Distributions
3.1 The binomial distribution
3.2 The hypergeometric distribution
3.3 The Poisson distribution
3.4 The normal probability density
3.5 Central limit theorem and the normal distribution
3.6 More on probability densities
3.6.1 The uniform and the beta densities
3.7 The Poisson process
3.8 The Q-Q plot and the chi-square test
3.9 The bivariate normal density
3.9.1 Additional material for joint random variables
Chapter 4. Real-World Applications of Probability
4.1 Fraud in a Canadian lottery
4.2 Bombs over London in World War II
4.3 Winning the lottery twice
4.4 Santa Claus and a baby whisperer
4.5 Birthdays and 500 Oldsmobiles
4.6 Cash Winfall lottery: a revenue model for stats geeks
4.7 Coupon collecting
4.8 Benford's law
4.9 What is casino credit worth?
4.10 Devil's card game: a psychological test
Chapter 5. Monte Carlo Simulation and Probability
5.1 Introduction
5.2 Simulation tools
5.2.1 Random number generators
5.2.2 Simulating from a nite range
5.2.3 Simulating a random permutation
5.2.4 Hit-and-miss method
5.2.5 Rejection sampling
5.3 Probability applications of simulation
5.3.1 Geometric probability problems
5.3.2 Almost-birthday problem
5.3.3 Consecutive numbers in lottery
5.3.4 Mississippi problem
5.3.5 Venice-53 lottery: what's in a number?
5.3.6 Kruskal's count and another card game
5.3.7 Randomized quick-sort algorithm
5.4 Bootstrap method in data analysis
5.5 Statistical analysis of simulation output
5.5.1 Variance reduction through importance sampling
Chapter 6. A Gentle Introduction to Markov Chains
6.1 Markov chain model
6.2 Absorbing Markov chains
6.3 The gambler's ruin problem
6.4 Long-run behavior of Markov chains
6.5 Markov chain Monte Carlo simulation
6.5.1 Metropolis{Hastings algorithm
6.5.2 Gibbs sampler
Solutions to Selected Problems
Index