Contents
Preface
Chapter 1. Combinatorics and Calculus for Probability
1.1 Factorials and binomial coefficients
1.2 Basic results from calculus
Chapter 2. Basics of Probability
2.1 Foundation of probability
2.2 The concept of conditional probability
2.3 The law of conditional probability
2.4 Bayesian probability
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
Appendix: Proofs for expected value and variance
Chapter 3. Useful Probability Distributions
3.1 The binomial and hypergeometric distributions
3.2 The Poisson distribution
3.3 The normal probability density
3.4 Central limit theorem and the normal distribution
3.5 The uniform and exponential probability densities
3.5.1 The uniform density function
3.5.2 The exponential density function
3.6 The bivariate normal density
3.7 The chi-square test
Appendix: Poisson and binomial probabilities
Chapter 4. Real-Life Examples of Poisson Probabilities
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
Chapter 5. Monte Carlo Simulation and Probability
5.1 Introduction
5.2 Simulation tools
5.2.1 Random number generators
5.2.2 Simulating a random number from a finite interval
5.2.3 Simulating a random integer from a finite range
5.2.4 Simulating a random permutation
5.2.5 Simulating a random point inside a circle
5.3 Applications of computer simulation
5.3.1 Geometric probability problems
5.3.2 Birthday problems
5.3.3 Lottery problem
5.3.4 The Mississippi problem
5.4 Statistical analysis of simulation output
Appendix: Python programs for simulation
Chapter 6. A Primer on 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
Solutions to Selected Problems
Index