Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 20H / i

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- Solution i

Solution

For a finite irreducible Markov chain with invariant distribution

π

, the mean recurrence time of state

i

is

E_{i} T_{i}^{+} = \frac{1}{π _{i}} .

The simple random walk on the hypercube has the uniform invariant distribution because the hypercube graph is regular, so every vertex has mass

2^{- n}

. Therefore the expected first positive return time to the initial vertex is

2^{n} .

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