1. (Problem 6.11) Consider the Markov chain shown below. Let us refer to a transition that results in a state with a higher (respectively, lower) index as a birth (respectively, death). Calculate the following quantities, assuming that when we start observing the chain, it is in steady state. 0.4 0.5 0.8 0.6 0.2 3 0.3 0.2 (a) For each state i, the probability that the current state is i. (b) The probability that the first transition we observe is a birth. (c) The probability that the first change of state we observe is a birth. (d) The conditional probability that the process was in state 2 before the first transition that we observe, given that this transition was a birth. (e) The conditional probability that the process was in state 2 before the first change of state that we observe, given that this change of state was a birth. (f) The conditional probability that he first observed transition is a birth given that it resulted in a change of state. (g) The conditional probability that the first observed transition leads to state 2, given that it resulted in a change of state