# 02fafin - 632 Introduction to Stochastic Processes Fall...

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Unformatted text preview: 632 Introduction to Stochastic Processes Fall 2002 Final Exam Instructions: Hand in all four problems. The points add up to 100. Show calculations and justify non-obvious statements for full credit. 1. (a) (15 pts) Suppose X n is a discrete-time Markov chain with state space S = { 1 , 2 } and transition matrix P = 1- α α β 1- β where α, β ∈ (0 , 1). Start the chain with its invariant distribution π . For positive integers m 3 > m 2 > m 1 > 0, calculate the probability P π X n = 1 for m 1 ≤ n ≤ m 2 , X n = 2 for m 2 < n ≤ m 3 . (b) (15 pts) Suppose X ( t ) is a continuous-time Markov chain with state space S = { 1 , 2 } and rate matrix Q =- μ μ ν- ν with μ,ν > 0. Let r > 0. Find the value of the limit lim t →∞ P 1 there are no jumps during time interval ( t,t + r ] . 2. (20 pts) Two transportation routes 1 and 2 operate independently of each other between points A and B. The system has 4 parameters, λ 1 , μ 1 , λ 2 , and μ 2 . For both i = 1 and 2, route i operates for an exponentially distributed...
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02fafin - 632 Introduction to Stochastic Processes Fall...

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