Consider the Markov chain in Fig. 7.22. Let us refer to a

QUESTION:

Consider the Markov chain in Fig. 7.22. Let us refer to a transition that results in a state with higher (respectively lower) index as birth (respectively death). Calculate the following quantities, assuming that when we start observing the chain, it is already in steady-state (f) The conditional probability that the first observed transition is a birth given that it resulted in a change of state. (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 the first change of state we observe is birth. (d) The conditional probability that the process was in state 2 before the first transition that we observe, given that this transtion was a birth. (e) The conditional probabilty that the process was in state 2 before the first change of state that we observe, given that this change of state was birth (g) The conditional probability that the first observed transition leads to state 2, given that it resulted in a change of state.