{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Homework_4

# Homework_4 - 1 Read pages 192 193 of the textbook 2 Read...

This preview shows pages 1–2. Sign up to view the full content.

1. Read pages 192, 193 of the textbook. 2. Read the remark on page 200 of the textbook. 3. Read Proposition 4.3. 4. [15 points] A stochastic process { X n , n = 0 , 1 , 2 , · · ·} with state space S = { 0 , 1 , 2 , · · · , l } is called a Markov chain if for any i, j, i 0 , i 1 , · · · , i n - 1 S and n 0, P { X n +1 = j | X n = i, X n - 1 = i n - 1 , · · · , X 1 = i 1 , X 0 = i 0 } = P { X 1 = j | X 0 = i } . a) Use this definition to prove that for the Markov chain defined as above, and for n > k 0 P { X n = j | X k = i, X k - 1 6 = i k - 1 , · · · , X 1 6 = i 1 , X 0 6 = i 0 } = P { X n - k = j | X 0 = i } . b) For the Markov chain defined as above and for n > k 0, does P { X n = j | X k 6 = i, X k - 1 = i k - 1 , · · · , X 1 = i 1 , X 0 = i 0 } = P { X n - k = j | X 0 6 = i } ? Prove it or disprove it. 5. [20 points] Consider a Markov chain with 4 states { 0 , 1 , 2 , 3 } . The transition proba- bility matrix takes the following form: 2 / 5 1 / 5 1 / 5 a 1 / 5 2 / 5 1 / 5 b 1 / 5 1 / 5 2 / 5 c 1 / 5 1 / 5 1 / 5 d Let A = { 0 , 1 } and B = { 2 , 3 } . a) What are the values of a, b, c, d ? b) Which states are recurrent? Which states are transient?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

Homework_4 - 1 Read pages 192 193 of the textbook 2 Read...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online