 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-2/26k/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-05

Put

r = 1 - α \in (0, 1)

. The variable

Y = X_{1} - θ

r y^{r - 1}

(0, 1)

, so

E Y^{j} = r / (r + j)

. Thus

E X_{1} = θ + \frac{r}{r + 1}, Var (X_{1}) = \frac{r}{( r + 2 ) ( r + 1 ) ^{2}} .

θ_{n} = \overline{X}_{n} - \frac{1 - α}{2 - α}, c (α) = - \frac{1 - α}{2 - α} .

Taking expectations verifies that its bias is zero.

 New to topics? Read the docs here!