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A time-inhomogeneous Markov jump process has state space {A, B; and the transition rate for switching between states equals 2r, regardless of the state currently occupied, where / is time. The process starts in state A at / = 0. (i) Calculate the probability that the process remains in state A until at least time s. [2] (ii) Show that the probability that the process is in state B at time 7, and that it is in the first visit to state B, is given by Taxes [3] (@) Sketch the probability function given in (ii). (b) Give an explanation of the shape of the probability function. (c) Calculate the time at which it is most likely that the process is in its first visit to state B. [6] [Total 1 1]A car manufacturer runs tests to investigate the fuel consumption of cars using a newly developed fuel additive. Sixteen cars of the same make and age are used, eight with the new additive and eight as controls. The results, in miles per gallon over a test track under regulated conditions, are as follows: Control 27.0 32.2 30.4 28.0 26.5 25.5 29.6 27.2 Additive 31.4 29.9 33.2 34.4 32.0 28.7 26.1 30.3 If /c is the mean number of miles per gallon achieved by cars in the control group, and # is the mean number of miles per gallon achieved by cars in the group with fuel additive, test: (i) Ho: HA- HC =0 vs H1 : HA-HC>0 (ii) Ho: HA- HC =6 vs H1 : HA - HC #6