 Solution

ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2015/1/a/ii/solution

M_{U} = U_{(m + 1)}

. Since

n = 2 m + 1

, its index is

r = (n + 1) /2

and

n - r + 1 = (n + 1) /2

. Substitute these into the expectation and variance formulas from part (i):

E [M_{U}] = \frac{1}{2}, Var (M_{U}) = \frac{1}{4 ( n + 2 )} .

M_{U}

has the symmetric beta distribution

Beta (m + 1, m + 1)

, explaining why its expectation is exactly the centre of the interval.

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