A student in a laboratory is required to observe a counter and to record the times at which the value on the counter changes. Whenever the counter reaches 3, the student stops the experiment, then restarts it with the counter set to zero. Four runs of the experiment are conducted, with the following results: Transition Time at which transition occurred 1st run 2nd run 3rd run 4th run 0 to 1 2.1 6.0 4.3 7.6 1 to 2 19.9 15.6 13.3 21.2 2 to 3 37.6 21.9 29.4 45.1 The times are measured from the beginning of the run. (i) Suppose that the number showing on the counter at time t, N(t), is to be modelled as a Poisson process. (a) Estimate the rate parameter, A, of the process. (b) If your model is accurate, what is P (N(t) = 1|N(0) = 0) ? (ii) Now suppose that a more general Markov jump process (MJP) is to be used as a model, with state space S {0,1,2,3}. (a) Estimate the transition rates 901: 912 and 923 of the model. (b) Assume this second model is accurate, and denote by p.,(t) the probability that the counter shows j at time t given that it showed i at time 0. Explain why it is the case that Poi(t) = 0.2e-a2pni(tu)du (c) Show that this equation is equivalent to the Kolmogorov Backward Equation for poi (t). Calculate poi(t). (iii) (a) Comment on any aspects of the data which might lead you to believe that the MJP model might provide a significantly better fit to the data than the Poisson process model. (b) Briefly indicate how you might test whether the fit of the MJP model is better than that of the Poisson process model.