Stat3021 Assignment 1
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Assignment 1 (for Stat3021)
1. The random variables ξ, ξ1 , ξ2 , . . . are independent and identically distributed with distribution P (ξ = 0) = 1/4 and P (ξ = j) = c/j for j = 1, 2, 3. Let X0 = 0 and Xn = max(ξ1 , . . . , ξn ) for n = 1, 2, . . ..
(a) What value must c take?
(b) Explain why {Xn , n = 0, 1, 2, . . .} is a Markov chain.
(c) Write down the transition matrix.
(d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc).
(e) Calculate P (Xn = 0).
(f) Calculate P (X4 = 3, X2 = 1|X1 = 3).
2. Consider a Markov chain {Xn }n≥0 having the following transition diagram:
1/2
1/4
1/2
1/4
1/2
For this chain, there are two recurrent classes R1 = {6, 7} and R2 = {1, 2, 5}, and one transient class R3 = {3, 4}.
(a) Find the period of state 3.
(b) Find f33 and f22 .
(c) Starting at state 3, find the probability that the chain is absorbed into R1 .
(d) Starting at state 3, find the mean absorbation time, i.e., the expected number of steps that the chain is absorbed into R1 or R2 .
Note: there are missing transition probabilities for this chain, but no impact for your solution.
3. A rat is put into the following maze:
The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n moves.
(a) Explain why {Xn } is a MC and find the transition matrix P .
(b) Explain why the chain is irreducible, aperiodic and positive recurrent.
(c) What is the limit of Pn ? Explain.
(d) Find the probability that the rat is in compartment 3 after two moves.
(e) In the long run, how many times that the rat enters in compartment 4 in 100 movements?
2023-03-15