A-LevelMathematicsProbability and statistics9709/61 · May/June 2024 · Q68 Marks
The diagram shows the graph of the probability density function, f, of a random variable X . The graph is a quarter circle entirely in the first quadrant with centre (0,0) and radius a, where a is a positive constant. Elsewhere f(x) = 0. [Figure X.X]
- (a)Show that a = 2/√π[2]
- (b)Show that f(x) = √(4/π - x²)[2]
- (c)Show that E(X) = 8/(3√π)[4]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.
- [M1] 1/4 π a^2 = 1
- [A1] a = 2/√π
- [M1] x²+y² = (2/√π)²
- [A1] f(x) = √(4/π - x²)
- [B1] ∫₀^a xf(x)dx with correct limits (accept limits 0 and a).
- [M1] -1/3 [ (a² - x²)^(3/2) ]
- [A1] Wholly correct integration and limits.
- [A1] 8/(3√π)
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