STAT 521 -- FINAL EXAM REVIEW SHEET
(The final exam will be ROUGHLY 80% new material and 20% from Exam 1
and Exam 2 material. Also study the review sheets for Exams 1 and 2.)
POST-TEST 2 MATERIAL:
Last Part of Chapter 5 material:
F. Generalizations of Poisson Process
1. Nonhomogeneous (nonstationary) Poisson Process
2. Finding the mean value function
3. Compound Poisson Process
4. Conditional (mixed) Poisson Process
I. Continuous-Time Markov Chains
A. Basic Definition
1. State Space and Index set
2. Markovian property
3. Meaning of Stationary Transition Probabilities
4. Distribution of the amount of time a state spends in a state before transitioning
5. Parameters (v_j and P_ij) of a continuous-time Markov Chain
B. Birth and Death Processes
1. Definition
2. Interarrival rates and departure rates
3. State space, v_j's and transition probabilities P_ij
4. Pure birth processes, Yule process
5. Linear Growth model with immigration
6. M/M/1 and M/M/s queueing models
7. Times Until transtions T_i and finding their expected values
8. Finding expected time to go from state k to state j
C. Transition Probability Function
1. Definition
2. Instantaneous transition rates q_ij
3. Relationship of q_ij to v_j and P_ij
4. Chapman-Kolmogorov Equations
5. Kolmogorov's Backward Equations
a. Kolmogorov's Backward Equations for a pure birth process
b. Kolmogorov's Backward Equations for a birth and death process
6. Kolmogorov's Forward Equations
7. Computing Transition Probability Matrix using approximation methods
D. Limiting probabilities
1. Definition and sufficient condition for existence
2. Interpretation of p_j
3. Setting up the balance equations for general chains
4. Meaning of stationary probabilities
5. Setting up the balance equations for a birth and death process
6. Formulas for limiting probabilities P_j in a birth and death process
7. Finding Limiting probabilities for various examples
II. Brownian Motion and Related Processes
A. Basic properties of Brownian Motion
1. Definition and relationship to symmetric random walk
2. Connection to normal distribution
3. Standard Brownian motion
4. Distribution of X(t) at a given t
5. Conditional Distribution of X(s), given X(t), where s < t
B. Hitting Time Results
1. Definition of hitting time and formula for its distribution
2. Distribution of the maximum X(s) over [0,t]
3. Gambler's Ruin connections
C. Variations on Brownian Motion
1. Brownian Motion with Drift
2. Geometric Brownian Motion
3. Distribution of maximum of a Brownian Motion Process with Drift
D. Stochastic Integration and White Noise
1. Two forms of the definition of the stochastic integral
2. Expected value and variance
3. White noise
E. Gaussian Processes
1. Definition
2. Brownian Motion as a Gaussian Process
a. Its Expected Value and Covariance Formulas
3. The Brownian Bridge
a. Its Conditional Expected Value and Covariance Formulas
4. Integrated Brownian Motion
a. Its Expected Value and Covariance Formulas
F. Stationary Processes
1. Definition
2. Simple examples
3. Random Telegraph Signal Process
a. Its Expected Value and Covariance Formulas
4. Definition of Weakly Stationary Processes
5. Property of Gaussian Processes that are weakly stationary
6. Ornstein-Uhlenbeck Process
a. Its Expected Value and Covariance Formulas
7. Nonsymmetric Random Telegraph Process
a. Its Expected Value and Covariance Formulas
8. Autoregressive Process
a. Its Expected Value and Covariance Formulas
9. Moving Average Process
a. Its Expected Value and Covariance Formulas