STAT 521 EXAM 2 REVIEW SHEET I. Leftover Material from Chapter 4 A. Branching Processes 1. Basic Assumptions of Process 2. Definition of {X_n} 3. Meaning of mu and sigma^2 4. E[X_n] and var[X_n] 5. Extinction Probability pi_0 a. Relationship to mu b. Finding pi_0 when mu > 1 II. Markov Chain Monte Carlo (MCMC) A. Purpose of Monte Carlo simulation 1. Law of Large Numbers 2. Situation when Normalizing Constant is Unknown B. Metropolis-Hastings Algorithm 1. Target distribution 2. Proposal distribution and q(i,j) 3. Acceptance ratio and transition probabilities 4. Time-reversibility and detailed-balance condition 5. What makes a good choice of proposal distribution? 6. Usefulness of symmetric proposal distribution 7. Acceptance rate C. Gibbs Sampler 1. When it is used? 2. Relationship to Metropolis-Hastings Algorithm 3. Full conditional distributions 4. Burn-in III. The Exponential Distribution A. Properties of the Exponential 1. Its pdf, cdf, mgf, rate, mean, variance 2. Finding exponential probabilities by hand B. Memoryless Property 1. Definition of memoryless property 2. Proof that the exponential is memoryless 3. Using memoryless property to simplify finding probabilities 4. Exponential as the UNIQUE continuous distribution that is memoryless C. More Exponential Properties 1. Distribution of sum of iid expon. r.v.'s with same rate 2. P(X1 < X2), where X1, X2 are independent expon. r.v.'s 3. Distribution of minimum of set of independent expon. r.v.'s 4. Independence of minimum and rank-order of expon. sample 5. Probability that a particular Xi is the minimum D. Sums of Exponential Random Variables 1. Definition of hypoexponential r.v. 2. Distribution of sum of independent expon. r.v.'s with different rates 3. Definition and pdf of Coxian r.v. 4. Finding probabilities using numerical integration IV. Counting Processes and Poisson Processes A. Definition and Properties of Counting Processes 1. Independent increments 2. Stationary Increments B. Poisson Process 1. o(h) notation 2. Definition of Poisson Process 3. Poisson Approximation of Binomial Probability 4. Distribution of Number of events in a fixed interval C. Interarrival and Waiting Times 1. Definition of Interarrival Times T1, T2, T3, ... 2. Distribution of Interarrival Times 3. Definition of Waiting Times S1, S2, S3, ... 4. Finding probabilities involving Interarrival and Waiting Times D. Types of Events 1. Processes tracking the numbers of Type I and Type II events 2. Distributions of N1(t) and N2(t) 3. Generalization to more than Two Types of Events E. Conditional Distributions of Arrival Times 1. Relationship of Joint Distribution of S1,...,Sn to Uniform distribution 2. Joint distribution of S1,...,S{n-1} given Sn = t