A-LevelMathematicsProbability and statistics9709/63 · May/June 2022 · Q79 Marks
The random variables X and W have probability density functions f and g defined as follows: f(x) = { p(a²-x²) for 0 ≤ x ≤ a, 0 otherwise, g(w) = { q(a²-w²) for -a ≤ w ≤ a, 0 otherwise, where a, p and q are constants.
- (a)(i)Write down the value of P(X ≥ 0).[1]
- (a)(ii)Write down the value of P(W ≥ 0).[1]
- (a)(iii)Write down an expression for q in terms of p only.[1]
- (b)Given that E(X) = 3, find the value of a.[6]
Mark scheme
Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.
- [B1] 1
- [B1] 1/2
- [B1] q = 1/2 p
- [M1] ∫[0,a] p(a²-x²) dx = 1
- [A1] p = 3/(2a³)
- [M1] ∫[0,a] x p(a²-x²) dx = 3
- [A1] p(a⁴/4) = 3
- [M1] (3/(2a³)) * (a⁴/4) = 3
- [A1] a = 8
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