Further Mathematics · AQA 7367 · Option S · Topic SB

Poisson distribution

The Poisson distribution for counts of random events, and tests about its mean.

  • 5 sub-topics
  • Taught in Year 12
  • 86 practice questions
  • 5 animated lessons
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What the specification says

Every sub-topic AQA can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. SB1The Poisson distribution and when to use itY12
  2. SB2Poisson probabilitiesY12
  3. SB3Mean, variance and standard deviationY12
  4. SB4Sums of independent Poisson variablesY12
  5. SB5Hypothesis test for the mean of a Poisson distributionY12

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 86 questions in this topic, free, no account needed.

Poisson and binomial distributions

Easy

The Poisson distribution · Year 12 · 1 mark

  1. X∼Po(3.0)X \sim \mathrm{Po}(3.0). What is P(X=0)\mathrm{P}(X = 0)?
    (1)

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  1. P(X=0)\mathrm{P}(X = 0)=e−λ{}= \mathrm{e}^{-\lambda}.CC: 0.04980.0498

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Poisson and binomial distributions

Medium

The Poisson distribution · Year 12 · 7 marks

Calls arrive at a help desk at random, at an average rate of 1.5 per hour.
  1. (a)
    Give two reasons why a Poisson distribution may be a suitable model for the number of calls in an hour.
    (2)
  2. (b)
    Find the probability that at least 3 calls arrive in a 3-hour period.
    (3)
  3. (c)
    Find the probability that at least one call arrives in each of three separate one-hour periods.
    (2)
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Part (a)2 marks

  1. Calls arrive singly and independently.
  2. Calls arrive at a constant average rate.

Part (b)3 marks

  1. Scale the rate to the interval.Y∼Po(4.5)Y \sim \mathrm{Po}(4.5)
  2. P(Y≥3)\mathrm{P}(Y \geq 3)=1−P(Y≤2){}= 1 - \mathrm{P}(Y \leq 2)
  3. =0.8264= 0.8264

Part (c)2 marks

  1. Independent hours.(1−e−1.5)3\left(1 - \mathrm{e}^{-1.5}\right)^3
  2. =0.4689= 0.4689

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Hypothesis testing

Hard

Hypothesis test for a Poisson mean · Year 12 · 6 marks

The number of faults on a production line has historically followed a Poisson distribution with mean 1.2 per week. After a change of supplier, 10 faults are recorded in 5 weeks.
  1. Test, at the 5% level of significance, whether the rate of faults has increased. State your hypotheses clearly.
    (6)
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  1. λ\lambda is the mean for 5 weeks.H0:λH_0: \lambda=6, H1:λ>6{}= 6, \ H_1: \lambda > 6
  2. X∼Po(6)X \sim \mathrm{Po}(6)
  3. P(X≥10)=0.0839\mathrm{P}(X \geq 10) = 0.0839
  4. Compare with the significance level.0.0839>0.050.0839 > 0.05
  5. A conclusion in context.Do not reject H₀: there is insufficient evidence that the rate has increased.

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