1. A machine can be in one of four states: 'running smoothly' (state 1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and 'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. • If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine to explode. If the machine is running but needing adjustment there is a 20% chance that an engineer will repair it so it is running smoothly the next day and otherwise it will remain in the same state for the next day. Taking X₂ to be the state of the machine on the morning of day i for i EN we get a Markov chain which models the state of the machine. (a) Write down the transition matrix for this Markov chain. (b) The factory manager is interested in the number of days of smooth running we expect in the lifetime of the machine assuming that it starts its life running smoothly. Express this question in the Markov chain terminology we have developed in this module. Which Theorem in the notes can we use to calculate this? (c) Calculate the expectation of the number of days of smooth running in the lifetime of the machine. (d) The following mathematical expression describes an event. Give a description of this event as you would express it to a non-mathematician: , . . .‚ T — 1} : X₂ = 1}| > |{n € {0, 1,..., T-1}: Xn = 2}| |{n € {0, 1,...,: where T min{n: Xn = 4}. =