 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-2/4f/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

1 = c \int_{0}^{3} \frac{d x}{1 + x ^{2}} = c arctan (3) = c \frac{π}{3},

c = \frac{3}{π} .

E X = \frac{3}{π} \int_{0}^{3} \frac{x}{1 + x ^{2}} d x = \frac{3 lo g 2}{π} .

The second moment is

E X^{2} = \frac{3}{π} \int_{0}^{3} \frac{x ^{2}}{1 + x ^{2}} d x = \frac{3 3}{π} - 1.

Therefore the variance is

var X = \frac{3 3}{π} - 1 - \frac{9 ( lo g 2 ) ^{2}}{π ^{2}} .

For the median

m

\frac{3}{π} arctan m = \frac{1}{2},

m = tan \frac{π}{6} = \frac{1}{3} .

 New to topics? Read the docs here!