Exponential and Poisson distributions — Edexcel A-Level Statistics
Test yourself on Exponential and Poisson distributions with PEARSON EDEXCEL A-Level practice questions.
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Exponential and Poisson distributions explained
A Poisson model counts the number of events in a fixed interval of time, length, area or volume.
Read the full explanation
It is appropriate when events occur singly, independently, at a constant mean rate λ, and the mean count is proportional to the interval size. In a real-world situation, identify the interval, check that events do not occur in pairs or clusters, that one event does not change the chance of another, and that the rate is stable. For example, radioactive emissions counted by a Geiger counter over 10 seconds may be modelled as Poisson if the source is steady and counts are independent. If events cluster or the rate varies, the Poisson model is not appropriate.
Your focus
- State the conditions under which a Poisson model is appropriate for a real-world count.
- Apply the Poisson assumptions to a given context and decide whether the model is suitable.
- Explain why a Poisson model is not appropriate when events cluster, occur in pairs, or the rate varies.
Exponential and Poisson distributions exam tips
Marking Points
- States that the Poisson model counts the number of events in a fixed interval of time, length, area or volume.
- Identifies the modelling assumptions: events occur singly, independently, at a constant mean rate λ, and the mean count is proportional to the interval size.
- Applies the assumptions to a real-world context, for example checking whether emissions, arrivals or defects occur singly and independently at a steady rate.
- Explains that if events cluster, occur in pairs, or the rate changes within the interval, the Poisson model is not appropriate.
- Uses the correct notation: if X ~ Poisson(λ), then P(X = r) = e^(−λ) λ^r / r! for r = 0, 1, 2, …
Examiner Tips
- 💡Read the context carefully and name the interval (time, length, area or volume) before deciding whether a Poisson model is appropriate.
- 💡Check each assumption in turn: singly, independently, constant rate, and mean proportional to interval size; state which assumption fails if the model is not appropriate.
- 💡When a question gives a rate per unit interval, scale λ correctly for the interval actually used, and keep λ positive.
Common Mistakes
- Misunderstanding: any count data can be modelled by a Poisson distribution. Correction: the events must occur singly, independently and at a constant rate; clustered or dependent events require a different model.
- Misunderstanding: the Poisson parameter λ is the probability of an event. Correction: λ is the mean number of events in the chosen interval, not a probability, and it must be positive.
- Misunderstanding: the interval size does not affect λ. Correction: the mean count is proportional to the interval size, so λ must be scaled when the interval changes.
- Misunderstanding: events may occur simultaneously. Correction: the model assumes events occur singly, so simultaneous or paired events violate the assumptions.