Weeks 1 and 2: Chapter 1 - Karlin and Taylor
Study in depth:
1. Fully understand section 1, especially conditional distributions
and conditional expectation.
2. Derivation of Poisson process. Section 2, Example 2.
3. Memorize and understand the definition of a Markov process, (c) page 29.
4. Know the generating function for a random sum. Page 12-13.
Problem Set #1:
Elementary Problems #1-7, Problems #1,7-11,26. Due Jan. 22.
Remark:
Suppose there are n female Med flies in an isolated part of Florida.
Sufficient sterile males are released so that the probability of a fertile
mating is reduced to p < 1. the females have offspring which are
independent with mean family size $nu$ and variance $tau sup 2$. This is
an example of Elementary Problem #4.
Week 3: Chapter 2 - Karlin and Taylor
Study in depth:
1. Work through examples in section 2 in detail.
2. Know definitions of accessible, communicate, aperiodic, recurrent,
transient, and irreducible.
3. Understand sections 3 and 4, but don't spend much time on "periodicity".
Pages 62-63 and Theorem 5.1 which it leads to are important.
4. Don't spend much time on Lemma 5.1 - (a) is dominated convergence,
(b) is monotone convergence.
Problem Set #2:
Elementary Problems #1,6,7, Problems #5,7,8. Due Feb. 16
Remark:
You should be able to develop the transition matrix from a physical
model as in examples A-F.
Week 4: Chapter 3 - Karlin and Taylor
Overall assignment: Sections 1-4 and section 7.
Study in depth:
1. Know at least Theorems 1.2 and 1.3 and understand Remarks 1.3 and 1.4
which modify the basic theorems for recurrent classes and periodic chains.
2. Know how to compute the stationary distribution in Theorem 1.3, that is,
understand examples on pages 86-87 and 92-94. The example on page 92
illustrates some computations that are very important in applied probability.
3. We will be omitting sections 5 and 6, but if you have a particular
interest in queuing problems, you might want to look at them.
4. The result in section 7 is an application of the technique developed
in section 4. Don't worry about the details of the computations in
section 7, but you should understand the method.
Problem Set #4:
Elementary Problems #2,4,7,10,12, Problems #1,4,5. Due March 2
Week 5: Chapter 4 - Karlin and Taylor
Overall assignment: Sections 1-4.
Study in depth:
1. Thoroughly understand the derivation of the Poisson process and Pure
Birth Process. Do not spend much time on the counter problem in section 3
2. Understand the result dealing with the order statistics and the Poisson
process, as given in Theorem 2.3, and the preceeding results on the waiting
time distribution for the Poisson.
Problem Set #5:
Elementary Problems #1,11,24, Problems #3,12. Due March 14