Stat 433: Homework

Homework 1 solutions

Read Chapter 3 up to page 112
III.1 Exercises (p 98): 3
III.1 Problems (p 99): 4
III.2 Exercises (p 102): 2 (If this were a video game, what would the "map" look like? Draw a picture.)
III.2 Problems (p 104): 1, 2
III.3 exercises (p 112): 1

Read carefully p 116 - 130 (Then read it again!) This will probably take about 3 hours.
II.5 Problems (p 94): 5
III.4 Exercises (p 127): 1, 3, 4, 5
III.4 Problems (p 130): 1, 7, 17

Read 136 - 147.
III.4 Exercises (p 127): 7, 9
III.4 Problems (p 130): 4, 6, 10
III.5 Exercises (p 148): 2, 7
III.5 Problems (p 150): 3, 5 (note the maximal inequality this problem mentions is on page 90.)
Markov chains that are also Martingales: This homework will walk you through the proof I gave in class of the probability that a random walk gets absorbed into state zero. Assume that X_t is a random walk over states 0 to n. Let p=1/3 be the probability of going up, and q=2/3 be the probability of going down. We will define Y = 2^X.
1. Prove that Y_t is a martingale.
2. Compute E(Y_t) given Y₀ = 2ⁱ.
3. Estimate the probability that Y_t has not been absorbed by time t.
4. How large does t have to be so that we are sure that the probability of not being absorbed is less than 2^^-n?
5. Estimate P(Y_t = 1) given Y₀ = 2ⁱ.
6. Does this match the formula given in the book?

Sorry the sample midterms were all messed up.
- 2004 (answers)
- Another midterm (.ps)
p 63 # 1.7 (exam level of difficulty)
The following are about random sums. Treat the last 3 as homework level difficult problems.
- p 77 # 3.1 (exam level of difficulty)
- p 78 # 3.3 (exam level of difficulty)
- p 78 problem 3.1, 3.3, 3.4
p 94. # 5.3 (look over the other martingale problems in this section. Be sure you think they are easy now! If not--try them again.)
p 99 exercise 1.5: (exam level of difficulty)
p 99 problem 1.1: (exam level of difficulty)
p 103 # 2.5
p 104 # 2.3
p 112 # 3.2
p 114 # 3.4
p 127 # 4.2 (exam level of difficulty)
p 129 # 4.8 (exam level of difficulty)
p 130 # 4.2
p 132 # 4.8 (exam level of difficulty)
p 148 # 5.8
p 150 # 5.1

NEW QUESTIONS ON BRANCHING

Read carefully 290-294 (I didn't cover all of this in class, but you should read carefully enough that you understand it well.)
Read 297-305
V.2: problems (p 287): 2.2, 2.4, 2.10, 2.11
V.3: exercises (p 294): 3.8
V.3: problems (p 295): 3.3, 3.8
V.4: exercises (p 308): 4.3
V.4: problems (p 309): 4.6, 4.8

Optional: Read section 6.6 if you have taken linear algerbra.
Read section 8.1: up to page 481.
Read all of 8.2
start reading 8.3
VI.2: problems (p 354): 2.4
VI.3: exercises (p 364): 3.2
VI.3: problems (p 365): 3.2
VIII.1: exercises (p 487): 1.2, 1.3, 1.4
VIII.1: problems (p 488): 1.1, 1.2, 1.3, 1.5
Answer the 3rd question posed in the article passed out in class (and here on the web.)

(NOTE: This homework will not be collected.)

Read all of 8.3.
Read parts of section 8.4 (p 508-514 and 514-521. Stop each subsection when your eyes glaze over.)
VIII.1: exercises (p 487): 1.7
VIII.1: problems (p 488): 1.7
VIII.2: exercises (p 496): 2.4, 2.5
VIII.2: problems (p 497): 2.1, 2.2, 2.4
VIII.3: exercises (p 505): 3.1
VIII.3: problems (p 506): 3.2, 3.3, 3.8
VIII.4: exercises (p 521): 4.4
VIII.4: problems (p 522): 4.1

Another sample without answers final.

You will be allowed one new cheat sheet along with the one you made for the midterm. (no calculators on the exam)

Some questions that might be useful for review.

Last modified: Thu May 8 17:34:57 EDT 2008