A-LevelMathematicsProbability and statistics9709/61 · May/June 2025 · Q511 Marks

The random variables W and X have the independent distributions Po(1.2) and Po(2.3) respectively.

(a)(i)Find P(3 < W+X≤5).[2]
(a)(ii)The random variable S is the sum of 100 independent values of W and 200 independent values of X. Use a suitable approximation to find P(S > 600).[6]
(b)The random variable Y has the distribution Po(λ), where λ > 0. It is given that 5/2 P(Y = 3) + P(Y = 4) = P(Y = 5). Find the value of λ.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] Any λ, allow one end error. Must see expression or terms.
[A1] SCB1 for unsupported answer of 0.537.
[B1] Use of N(1.2×100 + 2.3 × 200, ...) = N(580, ....) SOI give at early stage.
[M1] Var(S) = 1.2 × 100 + 2.3 × 200 = 580 SOI give at early stage.
[M1] 600.5-'580' / √'580' For standardising with their values. Allow with omitted or incorrect continuity correction.
[A1] = 0.851 Allow with omitted or incorrect continuity correction.
[M1] 1 – Ф('0.581') For area consistent with their working
[A1] = 0.197 (3sf)
[B1] (5/2)λ^3/3! + λ^4/4! = λ^5/5! Or (5/2)λ^3 + λ^4/4 = λ^5/20 Accept with or without each term multiplied by e^-^.
[M1] λ^2 - 5λ - 50 = 0 Legitimately obtain quadratic in λ.
[A1] λ = 10 Alone.

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