A-LevelMathematicsProbability and statistics9709/62 · May/June 2023 · Q25 Marks

The random variable W has a Poisson distribution. State the relationship between E(W) and Var(W).

(a)The random variable W has a Poisson distribution. State the relationship between E(W) and Var(W).[1]
(b)The random variable X has the distribution B(n, p). Jyothi wishes to use a Poisson distribution as an approximate distribution for X. Use the formulae for E(X) and Var(X) to explain why it is necessary for p to be close to 0 for this to be a reasonable approximation.[1]
(c)Given that Y has the distribution B(20000, 0.00007), use a Poisson distribution to calculate an estimate of P(Y > 2).[3]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[1] E(W) = Var(W). Allow 'they are the same' OE. Must be = not ≈ (and not both = and ≈). Condone E(W) =λ and Var(W)= λ.
[1] np ≈ np(1-p), hence 1 – p must be close to 1 OE. Must see formulae and q = 1 -p must be seen or implied and conclusion made.
[1] λ = 1.4 Seen.
[1] 1 – e⁻¹⁴(1 + 1.4 + 1.4²/2) or 1 – e⁻¹⁴(1 + 1.4 +0.98) or 1- (0.2466+0.3452+0.2417) Allow any λ; allow one end error. Expression must be seen (accept correct sigma notation).
[1] 0.167 (3 sf) or 0.166 Use of Binomial scores SCB1 for 0.167 or 0.166. No working: 0.167 [or 0.166] SC B1. Note: λ=1.4 and 0.167 with no working seen scores SC B1B1. Use of Normal scores B0M0.

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.