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The Chinese University of Hong Kong

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Course Examination 2nd Term, 2012-13

*I- IHte ~lC£ ~ii Course Code & Title: MATH4240 Stochastic Processes -------......_------......_-----------....._------...._-----.._------------..--------------------~ ..-----..-----­*M +* ~.

Time allowed hours 0 minutes

*~ Student I.D. No. Seat No. :

Answer all questions

1. (12 pts.) A gambler either wins or loses $1 with probability ~ and ~ in each bet. Suppose he has $2 to start with, and he will leave the game as soon as he reaches $0 or $4. What is the probability he wins the game eventually?

2. (12 pts.) A fair coin is tossed repeatedly, what is the expected number of tosses to have three consecutive heads?

3. (12 pts.) Let {Xn}~=o be a Markov chain. Let N(y) be the number oftimes the chain visits the state y, Ty the first time the chain visits y and Pxy ::;::: Px(Ty < (0). Derive the formulas for the following quantities in terms of Pxy and Pyy.

(a) Px(N(y);;:: m),

(b) EAN(y)).

4. (12 pts.) Let {Xn}~=o be a Markov chain, show that:

(a) If x is positive recurrent and x --t y, then y is also positive recurrent.

(b) If the chain is irreducible and has finitely many states, then all the states are positive recurrent.

5. (12 pts.) Suppose {En}~l are independent, identically distributed exponential random variables with parameter .A. Let Tn = 6 -+ ... + Em evaluate P(Tn ;;:: t).

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6. (16 pts.) Suppose buses and taxis come as Poisson processes with rates Al and A2 respectively. Answer the following:

(a) The expected time for two buses to come.

(b) The expected time for a bus or a taxi to come.

(c) The expected time for both bus and taxi to come.

(d) The probability that a taxi comes before a bus.

7. (12 pts.) A machine is in operation or failure according to a Markov jump process. In average it fails once in n days; if it fails, the average time to fix is m days.

(a) Find the transition probability Pxy(t) of the process.

(b) In the long run, what is the probability that the machine is in operation? Justify your answer.

8. (12 pts.) A collection of particles are giving new generations as a branching process, each one will split into two with probability p or will perish with probability q (l-p). The waiting time for a such change is an exponential distribution with rate A.

(a) Find the rate matrix to describe the process.

(b) If in addition there are new particles immigrate into the system as a Poisson process with rate Q, what is the new rate matrix?

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