Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 4 / b / iv

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- Solution iv

Solution

Apply the law of total expectation to the partition of a sample space given by the possible values of

R_{k}

. Since the conditional mean is affine,

r_{k + 1} = E [E [R_{k + 1} ∣ R_{k}]] = 1 + (1 - \frac{2}{n}) r_{k} .

Here

r_{0} = n

. This uses linearity of expectation and does not assert independence of successive urn counts.

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