Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 20H / b / i

Table of contents
- Solution i

Solution

The states

b, d

form a closed communicating class. Set

p_{n} = P (X_{n} = b ∣ X_{0} = d)

. Within this class,

p_{n + 1} = \frac{1}{4} p_{n} + \frac{2}{3} (1 - p_{n}) = \frac{2}{3} - \frac{5}{12} p_{n}, p_{0} = 0.

Its fixed point is

8/17

. Subtracting that fixed point and iterating gives the exact transition probability

p_{n} = \frac{8}{17} (1 - (- \frac{5}{12})^{n}), n \geq 0.

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