A random variable X has probability density function given by f(x) = { ax for 0 ≤ x ≤ b, { 0 otherwise, where a and b are constants.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [M1] Attempt triangle area or integrate f(x) (ignore limits) and = 1.
2. [A1] Correct integration. 1/2 x b x ab = 1.
3. [A1] Convincingly obtained. No errors seen.
4. [M1] Attempt to integrate xf (x). Ignore limits.
5. [A1] Correct integration and limits.
6. [A1] OE. E.g. ab³/3
7. [M1] OE. Integrate f(x) with limits 0 and their E(X) Condone a instead of 2/b² Or using areas, prob= 1/2 x 2b/3 x 2b/3 x ab
8. [A1FT] Correct integration and limits, FT their E(X). Need 2/b². Or prob= 1/2 x '2b/3' x '2/b² x 2b/3'
9. [A1] Must see correct expression after substituting limits. Convincingly obtained. No errors seen. Or expression correctly simplified.