Contents
1.9 Probability measures are continuous
2.1 Probability mass functions
2.3 Functions of discrete random variables
2.5 Conditional expectation and the partition theorem
3 Multivariate discrete distributions and independence
3.1 Bivariate discrete distributions
3.2 Expectation in the multivariate case
3.3 Independence of discrete random variables
4 Probability generating functions
4.2 Integer-valued random variables
4.4 Sums of independent random variables
5 Distribution functions and density functions
5.2 Examples of distribution functions
5.3 Continuous random variables
5.4 Some common density functions
5.5 Functions of random variables
5.6 Expectations of continuous random variables
6 Multivariate distributions and independence
6.1 Random vectors and independence
6.3 Marginal density functions and independence
6.4 Sums of continuous random variables
6.6 Conditional density functions
6.7 Expectations of continuous random variables
6.8 Bivariate normal distribution
7 Moments, and moment generating functions
7.4 Moment generating functions
8.2 Chebyshev’s inequality and the weak law
8.4 Large deviations and Cramér’s theorem
8.5 Convergence in distribution, and characteristic functions
9.2 A model for population growth
9.3 The generating-function method
9.5 The probability of extinction
10.1 One-dimensional random walks
10.3 Recurrence and transience of random walks
10.4 The Gambler’s Ruin Problem
11 Random processes in continuous time
11.1 Life at a telephone switchboard
11.3 Inter-arrival times and the exponential distribution
11.4 Population growth, and the simple birth process
11.5 Birth and death processes
12.4 Recurrence and transience
12.5 Random walks in one, two, and three dimensions
12.6 Hitting times and hitting probabilities
12.7 Stopping times and the strong Markov property
12.10 Convergence to equilibrium
Appendix A Elements of combinatorics