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A-LevelMathematicsProbability and statistics9709/61 · May/June 2020 · Q611 Marks

The length of time, T minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given by f(t) = { (3/4000)(20t - t²) 0 ≤ t ≤ 20, 0 otherwise.

  1. (a)Sketch the graph of y = f(t).[1]
  2. (b)Hence explain, without calculation, why E(T) = 10.[1]
  3. (c)Find Var(T).[3]
  4. (d)It is given that P(T < 10 + a) = p, where 0 < a < 10. Find P(10 – a < T < 10 + a) in terms of p.[2]
  5. (e)Find P(8 < T < 12).[3]
  6. (f)Give one reason why this model may be unrealistic.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

  1. [B1] 'Tails down' parabola only from x = 0 to 20 shown
  2. [B1] Symmetrical
  3. [M1] 3/4000 ∫(20t³ - t⁴)dx evaluated for E(T²)
  4. [M1] Var(T) = (3/4000)×160000-10²
  5. [A1] 20
  6. [M1] (p-0.5)×2 or 1-2(1-p)
  7. [A1] 2p-1
  8. [M1] 3/4000 ∫₁₂⁸ (20t-t²)dt
  9. [A1] [3/4000 * (10t² - t³/3)] evaluated from 8 to 12
  10. [A1] 37/125 or 0.296
  11. [B1] Does not allow times greater than 20 minutes

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About this question

Topic
Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 11 marks.
Source
A-Level Mathematics, May/June 2020 Paper 6 Variant 1, question 6.

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