Contents
1 Functions of random variables
1.1 Cumulative distribution technique
1.2 Sums of independent random variables
1.3 Sums of independent binomial variables
1.4 Distribution of the range of a random sample
1.5 Joint distributions
1.6 Practice problems
2 Generating functions
2.1 The probability generating function
2.2 Properties of the probability generating function
2.3 Sums and linear functions
2.4 The moment generating function
2.5 The cumulant generating function
2.6 Uniqueness and continuity
2.7 Practice problems
3 Convolutions
3.1 The discrete case
3.2 The continuous case
3.3 The generating function shortcut
3.4 Practice problems
4 Conditional expectation
4.1 Introduction
4.2 Computing expectations by conditioning
4.3 Computing probabilities by conditioning
4.4 Conditional variance
4.5 The moment generating function of a random sum
4.6 Practice problems
5 Compound distributions
5.1 Mean, variance and generating function
5.2 The compound Poisson distribution
5.3 Practice problems
6 Modes of stochastic convergence
6.1 The four definitions
6.2 The hierarchy
6.3 Every converse fails
6.4 Practice problems
7 Limit theorems
7.1 Introduction
7.2 The weak law of large numbers and the central limit theorem
7.3 Slutsky’s theorem
7.4 Other inequalities
7.5 Practice problems
8 Additional topics in probability
8.1 Stochastic processes
8.2 The Poisson process
8.3 Markov chains
8.4 Transient and recurrent states
8.5 Practice problems
9 Martingales
9.1 Definition and first examples
9.2 Martingale differences and Hoeffding’s inequality
9.3 Convergence
9.4 Practice problems
10 Simulation
10.1 The probability integral transform
10.2 The inverse transform method
10.3 Rejection sampling
10.4 Simulating the normal distribution
10.5 Practice problems