(a) The probability density function of the random variable X is given by f(x) = { kx(4-x) 0 ≤ x ≤ 2, { 0 otherwise, where k is a constant. (i) Show that k = 3/16. (ii) Find E(X).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [M1] Attempt integral f(x) and = 1. Ignore limits (must see a power increase for attempted integration).
2. [A1] k [4x^2/2 - x^3/3] limits 2,0 = 1 Correct integration and correct limits.
3. [A1] k * (16/3) = 1 [k = 3/16] OE AG Convincingly obtained. At least one interim step. No errors seen.
4. [M1] Attempt integral xf(x). Ignore limits. (must see a power increase for attempted integration). Condone missing k.
5. [A1] 3/16 [4x^3/3 - x^4/4] limits 2,0 Correct integration and correct limits. Condone missing k.
6. [A1] 5/4 Unsupported correct answer scores SC B2 only.
7. [B1] Symmetrical frequency density graph, 0 to 5, showing area 0.2 to left of a With a to the left of centre.
8. [B1] Either 0.2 between 5-a and 5-or 0.8 between 0 and 5-a Shown on graph or stated (5–a seen or implied). a must be non-numerical.
9. [B1] [P(2.5<Y<5-a)] = 0.3 Must be clearly final answer. a must be non-numerical.