Random processes and Markov Chain

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1. An athletic facility has 5 tennis courts. Pairs of players arrive at the courts and use a court for an exponentially distributed time with mean 40 minutes. Suppose a pair of players arrives and finds all courts busy and k other pairs waiting in queue. What is the expected waiting time (in minutes) to get a court? *

40*k

40*(k+1)

8*k

8*(k+1)

2. In a twenty over IPL match (i.e., 120 deliveries), the probability of a wicket falling of a given delivery is 0.05. Assume that the wicket falling on each delivery is independent of others.

2 a) What is the probability of the team batting first to lose exactly 1 wicket at the end of their innings? *

0.0134

0.0234

0.591

0.0100

2 b) For the team batting first, what is the expected number of wickets they lose at the end of their innings? *

5.5

2 c) What is the probability of the team batting first to be all out, i.e., lose all their wickets inside 20 overs? (There are at least two ways to obtain the desired probability. For fun, obtain the answer via both ways and check if they are indeed equal). *

0.0942

0.9431

0.0786

0.9213

3. An absent-minded professor has two umbrellas that she uses when commuting from home to office and back. If it rains and an umbrella is available at her location, she takes it. Else, she gets wet. If it is not raining, she always forgets to take an umbrella. Suppose that it rains with probability p each time she commutes, independent of other times. What is the steady-state probability that she gets wet during a commute? *

(1-p)/(3-p)

p(1-p)/(3-p)

p+(1-p)/(3-p)

4. The probability transition matrix of a Markov chain on 7 states is given below, where P_{ij} is the probability of transitioning from state i to state j in one step and × is a non-zero probability.

4 a) Identify the transient states *

4 b) Identify the non-periodic recurrent class(es) *

4 c) Identify the periodic recurrent class(es) *

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