Reference no: EM132852366

1. In a certain town up in the mountains there are no bears at time t = 0. Brown bears and grizzly bears arrive according to two independent Poisson processes with respective arrival rates β and γ. Compute the probability that the first bear arriving in town is a brown bear. Suppose that the first bear arriving in town is a grizzly bear. How long does it take on average before this bear arrives? Give a reason for your answer. Find the probability that between two consecutive brown bears there arrive exactly r grizzly bears. Justify your answer.

2. Computers in the lab fail, on average, three times a day, according to the random arrival times of a Poisson process. Last week, 15 computers failed. Find the expected time of the last failure, and give an approximate time of day when the last failure occurred.

3. Consider a continuous-time Markov chain (Xt)t>=0 with state space S = {1, 2, 3, 4}. From state 1 the process is equally likely to either transition to 2 or 4; from state 2 and 3 the process is equally likely to transition to all other states; and from state 4 the process always transitions to state 1. Suppose further that the process spends on average square root(i) units of time in each state i in S. Provide the transition probability matrix P˜ for the embedded chain. Provide the holding time parameters qi . Provide the generator matrix Q. Justify your answer. Draw the transition rate graph.