Consider the Markov chain below: a) Suppose the chain is started in state i and goes through n transitions; let vi(n) be the expected number of transitions (out of the total of n) until the chain...

a) Suppose the chain is started in state i and goes through n transitions; let vi(n) be the expected number of transitions (out of the total of n) until the chain enters the trapping state, state 1. Find an expression for v(n) = (v₁(n), v₂(n), v₃(n))^T
in terms of v(n₁) (take v₁(n) = 0 for all n).

b) Solve numerically for lim_n→1
v(n). Interpret the meaning of the elements v_i
in the solution of (4.32).

c) Give a direct argument why (4.32) provides the solution directly to the expected time from each state to enter the trapping state.