Poisson distribution
Objective. Model event counts and combine independent Poisson variables.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
The Poisson distribution models the number of random events occurring in a fixed interval of time or space, given a known constant average rate λ, with probability P(X=r) = e^(−λ)λʳ/r!. A key property is that the sum of two independent Poisson variables with means λ₁ and λ₂ is itself Poisson with mean λ₁+λ₂, allowing combined events to be modelled easily. For example, if calls to a helpline arrive at a rate of 3 per hour, the number of calls in two hours follows a Poisson distribution with mean 6.
Worked example
Accidents on a stretch of road occur at a Poisson rate of 2 per week. Find the probability that exactly 3 accidents occur in a given week, to 4 decimal places.
- 1Here λ = 2 and r = 3
- 2Poisson formula: P(X=3) = e^(−2) × 2³ / 3!
- 3Compute: 2³=8, 3!=6, so P(X=3) = 8e^(−2)/6 = (4/3)e^(−2)
- 4Numerically, e^(−2) ≈ 0.1353, so P(X=3) ≈ (4/3)(0.1353) ≈ 0.1804
Answer: P(X=3) ≈ 0.1804
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Practice
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Start practisingCommon mistake. Students use the Poisson distribution for situations with a fixed finite number of trials and constant probability, which is actually binomial, confusing the two. Poisson applies to rare, independent events over a continuous interval with no fixed upper bound on counts.
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