solve the questions in the attachment.

7 A time-inhomogeneous Markov jump process has state space (A, B) and the transition rate for style switching between states equals 2t , regardless of the state currently occupied, where t is time. The process starts in state A at t = 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 T2 xe-T [3] (iii) (a) 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 11] 8 An illness-death model has three states: style 1 = healthy 2 = sick 3 = dead Let py denote the probability that a life in state / at age x is in state / at age x +t and let w+ denote the force of transition from state / to state / at age x +t. (i) Draw and label a diagram showing the three states and the transition intensities between them. [2] (ii) Show, from first principles, that in this illness-death model: ( PX = 1 Px Matt - 1 Px Hatt - ( Px Hxtt [6] [Total 8]