Book:Geoffrey Grimmett/Probability: An Introduction

Subject Matter

 * Probability Theory

Basic Probability

 * Events and probabilities
 * Experiments with chance
 * Outcomes and events
 * Probabilities
 * Probability spaces
 * Discrete sample spaces
 * Conditional probabilities
 * Independent events
 * The partition theorem
 * Probability measures are continuous


 * Discrete random variables
 * Probability mass functions
 * Examples
 * Functions of discrete random variables
 * Expectation
 * Conditional expectation and the partition theorem


 * Multivariate discrete distributions and independence
 * Bivariate discrete distributions
 * Expectation in the multivariate case
 * Independence of discrete random variables
 * Sums of random variables


 * Probability generating functions
 * Generating functions
 * Integer-valued random variables
 * Moments
 * Sums of independent random variables


 * Distribution functions and density functions
 * Distribution functions
 * Examples of distribution functions
 * Continuous random variables
 * Some common density functions
 * Functions of random variables
 * Expectations of continuous random variables

Further Probability

 * Multivariate distributions and independence
 * Random vectors and independence
 * Joint density functions
 * Marginal density functions and independence
 * Sums of continuous random variables
 * Changes of variables
 * Conditional density functions
 * Expectations of continuous random variables
 * Conditional expectation and the bivariate normal distribution


 * Moments, and moment generating functions
 * A general note
 * Moments
 * Variance and covariance
 * Moment generating functions
 * Characteristic functions


 * The two main limit theorems
 * The law of averages
 * Chebyshev's inequality and the weak law
 * The central limit theorem
 * Convergence in distribution, and characteristic functions

Random Processes

 * Branching processes
 * Random processes
 * A model for population growth
 * The generating-function method
 * An example
 * The probability of extinction


 * Random walks
 * One-dimensional random walks
 * Transition probabilities
 * Recurrence and transience in random walks
 * The Gambler's Ruin problem


 * Random processes in continuous time
 * Life at a telephone exchange
 * Poisson processes
 * Inter-arrival times and the exponential distribution
 * Population growth and the simple birth process
 * Birth and death processes
 * A simple queueing model

Appendix

 * Difference equations