1. Suppose that shocks occur according to a Poisson process with rate A> 0. Also suppose that each shock independently causes the system to fail with probability 0 < p < 1. Let N denote the number of shocks that it takes for the system to fail and let T denote the time of the failure.
(a) FindP{T>tNrrn}.
(b) FindP{NrrnT=t}.
(c) Describe how the results from part (b) could have alternatively been determined by considering an appropriate decomposition of the original Poisson process.
2. Let {N(t)}>0 be a nonhomogeneous Poisson process with intensity function A(t) > 0. Then the mean value function 4u(t) is given by ji(t) j A(x) dx. Recall that, for 0
(a) Find E[N(s)N(t)] for 0 < s < t.
(b) Find the covariance of N(s) and N(t) for 0
(c) Find the correlation between N(s) and N(t) for 0
5. Consider two urns, each of which contains ri-i balls. Initially, the first urn contains w white balls and ri-i ? w black balls, where 0
(a) Find the transition probabilities of the Markov chain.
(b) Find the stationary distribution of the Markov chain.
(c) Describe how the results from part (b) could have been determined by considering how the balls would be distributed between the two urns after the Markov chain has reached stationarity.