The random variable X has probability density function, f, given by f(x) = { 1/x² a < x < b, 0 otherwise, where a and b are positive constants.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [M1] Attempt to integrate xf(x) from a to b. ∫(from a to b) x * (1/x²) dx
2. [A1] [ln x] (from a to b) or ln b - ln a
3. [A1] Must see both statements. No errors seen. ln(b/a) = ln 2 or ln b - ln a = ln 2 => b = 2a (AG)
4. [M1] Attempt to integrate f(x) and equate to 1. Ignore limits. ∫(from a to b) (1/x²) dx = 1
5. [A1] Integrate with correct limits and substitute b = 2a. [-1/x] (from a to b) = 1 or -1/b + 1/a = 1 => -1/(2a) + 1/a = 1
6. [A1] Obtain convincingly (at least one step from previous answer), no errors seen (ignore a = 0). 1/(2a) = 1 => a = 1/2 (AG)
7. [M1] Attempt integrate f(x), with correct limits, and equate to 1/2. ∫(from a to m) (1/x²) dx = 1/2
8. [A1] oe. Correct equation after substituting limits. [-1/x] (from 0.5 to m) = 1/2 or -1/m + 1/0.5 = 1/2 or -1/m + 2 = 1/2
9. [A1] m = 2/3 or 0.667 (3 sf)