Basic Probability: What Every Math Student Should Know

✍ Scribed by Henk Tijms

• Publisher: World Scientific Publishing
• Year: 2021
• Tongue: English
• Leaves: 184
• Edition: 2
• Category: Library

✦ Synopsis

The second edition represents an ongoing effort to make probability accessible to students in a wide range of fields such as mathematics, statistics and data science, engineering, computer science, and business analytics. The book is written for those learning about probability for the first time. Revised and updated, the book is aimed specifically at statistics and data science students who need a solid introduction to the basics of probability. While retaining its focus on basic probability, including Bayesian probability and the interface between probability and computer simulation, this edition's significant revisions are as follows: Many extra motivational examples and problems New material on Bayesian probability, including two famous court cases New sections on real-world applications of the Poisson distribution New sections on generating functions and the bivariate normal density New chapter on Markov chains, including Markov chain Monte Carlo simulation The approach followed in the book is to develop probabilistic intuition before diving into details. The best way to learn probability is by practising on a lot of problems. Many instructive problems together with problem-solving strategies are given. Answers to all problems and worked-out solutions to selected problems are also provided. Henk Tijms is the author of several textbooks in the area of applied probability. In 2008, he had received the prestigious INFORMS Expository Writing Award for his work. He is active in popularizing probability at Dutch high schools.

✦ Table of Contents

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

✦ Subjects

Probability; Probabilities; Bayesian Probability; Poisson Distribution; Markov Chains; Monte Carlo Simulation; Generating Functions; Bivariate Normal Density