 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2011/m/m5/a/vi/solution

X_{2}

is uniform on

{0, \dots, N - 1}

. Conditional on

X_{2} = k \geq 1

, the mean number removed is

(k + 1) /2

; conditional on

X_{2} = 0

, it is zero. The law of total expectation therefore gives

E Y_{2} = \frac{1}{N} k = 1 \sum N - 1 \frac{k + 1}{2} = \frac{( N - 1 ) ( N + 2 )}{4 N} .

For

N = 1

, the sum is empty and the same formula gives zero.

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