STAT 521 EXAM 1 REVIEW SHEET
I. Review of Probability Theory
A. Experiments, Sample Space, Outcomes, Events
1. Union and Intersection
2. Mutually Exclusive Events
3. Complement
B. Probabilities of Events
1. Three Fundamental Axioms
2. Equally-likely-outcomes case
3. Basic Probability Rules
a. Complement Rule
b. Additive Rule, Extended Additive Rule, Inclusion-Exclusion
C. Conditional Probability
1. Definition of Conditional Probability
2. Multiplicative Rule
3. Independent Events
4. Pairwise Independence and Joint independence
D. Bayes' Formula
1. With Two Events
2. With More than Two Events
E. Random Variables
1. Discrete and Continuous
2. cdf, pmf, pdf
3. Expected Value and Variance
4. Moment Generating Functions
5. Common Probability Distributions
6. Joint Distributions of multivariate r.v.'s
a. Joint cdf, joint pmf/pdf, marginal pmf/pdf
b. Independent random variables
c. Variances and covariances and their properties
d. Independence and mgf's and related distributional facts
e. Markov's inequality, Chebyshev's inequality, LLN, CLT
F. Stochastic Processes
1. Basic definition
2. Index set, state space
3. Discrete-Time and Continuous-Time Processes
G. Conditional Distributions
1. Conditional pmf/pdf, conditional expectation
2. Formula for Iterated Expectation
3. Formula for Iterated Variance
4. Computing Probabilities by Conditioning
5. Computing Expected Values by Conditioning
II. Discrete-Time Markov Chains
A. Markovian Property
1. Definition and Implications
B. (One-step) Transition Probabilities
C. Chapman-Kolmogorov Equations and Related Topics
1. Getting n-step transition probabilities
2. Defining a special set of states to be an absorbing state A
3. Calculating Unconditional Probabilities
D. Classification of States
1. Accessibility and communication
2. Properties of communication
3. Classes of states and irreducibility
4. Absorbing states
5. Recurrent and transient states
6. Recurrence as a class property, and implications
7. Random Walk example
E. Long-Run Proportions and Limiting Probabilities
1. Positive and null recurrence
2. Long-Run Proportions pi_j and relationship to m_j
3. Positive recurrence as a class property
4. Finding long-run proportions
5. Long-Run Proportions as stationary probabilities
6. Limiting Probabilities
7. Periodic and aperiodic
8. Gambler's Ruin example
a. Formula for probability of gambler reaching goal
F. Mean Time Spent in Transient States
1. P_T matrix for transitions between transient states
2. s_ij values and matrix S
3. f_ij probability and relationship with s_ij