 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2009/m/m2/c/ii/solution

State one is left after one step and never revisited. The eventual probability mass in the closed class

{4, 5}

w = (1 - p) λ_{1} + λ_{4} + λ_{5} .

Within this class the stationary distribution is

(b / (a + b), a / (a + b))

, because the balance equation is

a π_{4} = b π_{5}

. The two-state recurrence shows convergence from either starting state, since

∣1 - a - b ∣ < 1

. The other closed class never reaches state five. Therefore

n \to \infty lim P (X_{n} = 5) = \frac{a}{a + b} ((1 - p) λ_{1} + λ_{4} + λ_{5}) .

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