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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-2/18h/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 18H / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

The self-loop at zero makes the irreducible chain aperiodic. The convergence theorem for irreducible, aperiodic, positive recurrent countable-state chains gives

P (X_{n} = i) ⟶ π_{i} .

(1)

Because

X

and

Y

are independent,

P (X_{n} = 0, Y_{n} = 1) = P (X_{n} = 0) P (Y_{n} = 1) ⟶ π_{0} π_{1} .

(2)

Therefore

n \to \infty lim P (X_{n} = 0, Y_{n} = 1) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} .

(3)

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