3.2 Markov Chains - First Step Analysis Exercises Solutions.pdf - THE CHINESE UNIVERSITY OF HONG KONG Department of Statistics STAT3007 Introduction to

# 3.2 Markov Chains - First Step Analysis Exercises Solutions.pdf

• Homework Help
• 2

This preview shows page 1 - 2 out of 2 pages.

THE CHINESE UNIVERSITY OF HONG KONG Department of Statistics STAT3007: Introduction to Stochastic Processes Markov Chains - First Step Analysis Exercises Solutions 1. (Slide 7 of the “Markov Chains - First Step Analysis” notes) P ( T > k | X 0 = 1) = P ( X 1 6 = 0 , 2 , X 2 6 = 0 , 2 , . . . , X k 6 = 0 , 2 | X 0 = 1) = P ( X 1 = 1 , X 2 = 1 , . . . , X k = 1 | X 0 = 1) = p 11 × p 11 × · · · × p 11 ( k times) = β k 2. (Exercise 3.4.2 in Pinsky and Karlin) Recall u 10 := P ( X T = 0 | X 0 = 1) and v 1 := E [ T | X 0 = 1], where T is absorption time. Using standard first step analysis, we arrive at these equations u 10 = 1 × 0 . 1 + u 10 × 0 . 6 + 0 × 0 . 3 v 1 = 1 × 0 . 1 + (1 + v 1 ) × 0 . 6 + 1 × 0 . 1 and these have solutions (a) u 10 = 1 / 4. (b) v 1 = 5 / 2. 3. (Exercise 3.4.4 in Pinsky and Karlin) We follow the hint. We want to find the mean number of tosses, so let’s make this the mean time to absorption. Let the states be { 0 , 1 , 2 } , so X n is the running total of successive heads. Let State 2 be the absorption state, so that once we have 2 successive heads, we stop playing the game and stay in State 2 forever. Say we’re in State 0. We move to State 1 w.p. 0.5 (we toss a Head) and stay in

• Fall '14

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern