The Poisson distribution as a model

The formula

X ~ Po(λ) has P(X = r) = e^-λ λ^r / r! for r = 0, 1, 2, ... with no upper limit. λ is the mean number of events in the interval. Note 0! = 1, so P(X = 0) = e^-λ.

Conditions, in context

A Poisson model is suitable when events occur:

• singly (one at a time, never two at exactly the same moment);
• independently of each other;
• randomly;
• at a constant average rate (constant mean number per interval).

Examiners stress that the condition must be written in context: 'customers enter the shop singly and independently', not just 'events are independent'. A 'constant probability' is not a Poisson condition: say constant rate.

Criticising a model

Look for a reason the rate might not be constant (more customers at lunchtime), events that come in groups (a family arriving together breaks 'singly'), or events that affect each other (one breakdown causing another).

Worked example

Calls arrive at a helpline at a mean rate of 2.5 per 10 minutes. Find the probability of exactly 3 calls in a 10-minute period.

1. X ~ Po(2.5), assuming calls arrive singly, independently and at a constant rate.
2. P(X = 3) = e^-2.5 × 2.5³ / 3!
3. = 0.08208 × 15.625 / 6

Answer: P(X = 3) = 0.2138

Check yourself

1. X ~ Po(2.5). Find P(X = 3). Give your answer to 4 decimal places.
   Show answer

   0.2138. e^-2.5 × 2.5³ / 6 = 0.2138.

2. Flaws in a roll of cloth occur at a mean rate of 4.2 per roll. Find the probability that a roll has no flaws. Give your answer to 4 decimal places.
   Show answer

   0.0150. P(X = 0) = e^-4.2 = 0.0150.

3. X ~ Po(1.8). Find P(X ≤ 2). Give your answer to 4 decimal places.
   Show answer

   0.7306. e^-1.8(1 + 1.8 + 1.8²/2) = 0.16530 × 4.42 = 0.7306.

4. X ~ Po(λ) and P(X = 1) = P(X = 2). Find λ.
   Show answer

   2. e^-λλ = e^-λλ²/2, so 1 = λ/2 and λ = 2.

5. Which is a condition for a Poisson model of the number of cars passing a checkpoint per minute?
   • Each car has a constant probability of passing
   • There is a fixed number of cars
   • Cars pass at a constant average rate
   • Cars pass in groups
   Show answer

   Cars pass at a constant average rate. Poisson needs a constant rate of events; examiners do not accept 'constant probability'.

6. Which would be least well modelled by a Poisson distribution?
   • The number of people arriving at a cinema per minute, when most come in couples or families
   • The number of typing errors per page of a long report
   • The number of radioactive decays per second from a sample
   • The number of emails arriving per hour during a working day
   Show answer

   The number of people arriving at a cinema per minute, when most come in couples or families. People arriving in groups break the 'singly' condition.

7. A model says customers enter a shop at random at a mean rate of 12 per hour. Which comment best criticises it?
   • Customers cannot enter at random
   • 12 is too large for a Poisson mean
   • The rate is likely to be higher at lunchtime, so it may not be constant
   • The number of customers is not a whole number
   Show answer

   The rate is likely to be higher at lunchtime, so it may not be constant. A changing rate through the day is a contextual reason to doubt the model.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S2
• Pearson Edexcel IAL Statistics S2 (WST02) examiners' report, October 2024
• Pearson Edexcel IAL Statistics S2 (WST02) mark scheme, June 2025 (copy hosted by Physics and Maths Tutor)

Last updated 8 November 2026. Spotted a mistake? Tell us.