A-LevelMathematicsProbability and statistics9709/61 · May/June 2022 · Q69 Marks
A random variable X has probability density function f. The graph of f(x) is a straight line segment parallel to the x-axis from x = 0 to x = a, where a is a positive constant.
- (a)State, in terms of a, the median of X.[1]
- (b)Find P(X > ³/₄a).[1]
- (c)Show that Var(X) = ¹/₁₂a².[5]
- (d)Given that P(X < b) = p, where 0 < b < ¹/₂a, find P(¹/₃b < X < a – ¹/₃b) in terms of p.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.
- [B1] a/2
- [B1] 1/4
- [B1] f(x) = 1/a SOI (may be seen in part (a) or part (b))
- [B1] E(X) = a/2 SOI
- [M1] ∫₀ᵃ x²(1/a)dx Attempt integrate their f(x) × x² with correct limits
- [A1] [x³/3a]₀ᵃ = a²/3
- [A1] a²/3 - (a/2)² or a²/3 - a²/4 [= a²/12 AG] Must see previous line and answer No errors seen
- [M1] (1/a) × b = p => b/a = p SOI (could be on a diagram) OR by integration: prob = 1-(2/3)(b/a)
- [A1] 1 - 2p/3
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