# The Poisson distribution

> CIE A-Level Mathematics · 9709
> Source: https://www.owlsprep.com/study/cie-9709-u4-the-poisson-distribution/

This module covers Poisson distribution properties, valid use conditions, probability formula, cumulative table usage, Poisson approximation to Binomial, and standard exam problem workflows.

**Prerequisites:** [Basic discrete probability distributions](https://www.owlsprep.com/study/cie-9709-u4-discrete-probability-distributions/); [Binomial distribution properties and calculations](https://www.owlsprep.com/study/cie-9709-u4-the-binomial-distribution/)

## Learning objectives

- Recall the 4 required conditions for a scenario to be validly modelled by a Poisson distribution
- Calculate individual Poisson probabilities using the standard formula and cumulative probabilities via official CIE tables
- Solve real-world problems involving rate parameters for events occurring over fixed intervals
- Apply the Poisson approximation to the Binomial distribution for large n and small p scenarios

## How often is this tested?

Based on OwlsPrep's analysis of official CIE 9709 S2 past papers (2016–2025): **The Poisson distribution** appears **96 times** in the last 10 years — **22.4%** of all S2 questions (96 of 428).

Most-tested forms: Poisson distribution: probability, mean, variance (80), Poisson as approximation to binomial (large n, small p) (34).

## Core Definition and Valid Use Conditions

**Poisson Distribution** — A discrete probability distribution that models the number of independent events occurring at a constant average rate in a fixed interval of time, space, or volume

*Notation:* X ~ Po(λ)

*Example:* Number of customer arrivals at a shop per hour, with an average of 4 arrivals per hour

A scenario can only be modelled with a Poisson distribution if all 4 of these conditions are met: events are fully independent of each other, events cannot occur simultaneously, the average rate of events is constant across the interval, and the probability of more than one event in a tiny sub-interval is negligible.

> **Memory Hook**
>
> PISR: Poisson requires events to be Independent, Single, at a constant average Rate

**Worked example:** State whether each scenario can be modelled by a Poisson distribution: 1) Number of 6s rolled in 20 independent die rolls, 2) Number of potholes per 1km stretch of road across a highway network with consistent maintenance.

1. Evaluate scenario 1: This counts successes in a fixed number of independent trials, which is a Binomial distribution, not Poisson, as there is no interval of space/time with a constant event rate.
2. Evaluate scenario 2: Potholes occur independently, one at a time, with a constant average rate per km, so it fully satisfies all 4 Poisson conditions.

**Check your understanding**

Confirm your understanding of Poisson conditions

1. Which of the following scenarios is NOT suitable for a Poisson model?

   - Number of emails received per day
   - Number of goals scored in a football match
   - Number of heads in 100 coin flips
   - Number of typos per textbook page

   *Why:* This is a Binomial scenario with fixed number of trials, not events over an interval.

## Poisson Probability Formula and Basic Calculations

$$P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!} \quad \text{for } r = 0, 1, 2, 3, ...$$

A unique property of the Poisson distribution is that the mean (expected value) is exactly equal to the variance, both equal to the rate parameter λ. This is a frequently tested point in CIE exam questions.

**Worked example:** Given X ~ Po(2.5), calculate P(X = 3) to 3 significant figures.

1. Identify the parameters: λ = 2.5, r = 3
2. $$P(X=3) = \frac{e^{-2.5} \times (2.5)^3}{3!}$$
3. Calculate intermediate values: e^{-2.5} ≈ 0.082085, 2.5^3 = 15.625, 3! = 6
4. $$P(X=3) = \frac{0.082085 \times 15.625}{6} ≈ 0.214$$

> **tip**
>
> Never round intermediate values to fewer than 4 significant figures, as CIE will deduct marks for final answers that are off due to early rounding.

## Cumulative Poisson Probability Table Usage

CIE provides official cumulative Poisson tables in the exam formula booklet, which give pre-calculated values of P(X ≤ x) for common λ values. These eliminate the need for repeated formula calculations for large r values.

**Exam command terms**

CIE uses standard phrasing for cumulative Poisson questions that you must interpret correctly to avoid off-by-one errors:

- **At most k** — Directly use P(X ≤ k) from the cumulative table

- **More than k** — Calculate 1 - P(X ≤ k)

- **Between a and b inclusive** — Calculate P(X ≤ b) - P(X ≤ a-1)

**Worked example:** If X ~ Po(3.2), use cumulative Poisson tables to find P(1 < X ≤ 4).

1. Rewrite the inequality to match cumulative table format: P(1 < X ≤ 4) = P(X ≤ 4) - P(X ≤ 1)
2. Look up λ = 3.2 in the cumulative table: P(X ≤ 4) = 0.7806, P(X ≤ 1) = 0.1712
3. Subtract the two values to get the final result: 0.7806 - 0.1712 = 0.609

## Poisson Approximation to the Binomial Distribution

**Derivation:** Show that Bin(n,p) can be approximated by Po(np) under specific conditions

*Starting from:* Binomial probability formula: P(X=r) = \binom{n}{r} p^r (1-p)^{n-r}

1. When n is very large (n ≥ 50) and p is very small (p < 0.1), set λ = np
2. As n → ∞, \binom{n}{r} ≈ \frac{n^r}{r!} and (1-p)^{n-r} ≈ e^{-np}
3. Substituting these approximations recovers the standard Poisson probability formula

*Conclusion:* Bin(n,p) ≈ Po(np) for large n, small p, and λ = np < 10

**Worked example:** A factory produces 1000 components, each with a 0.002 independent probability of being defective. Use Poisson approximation to find the probability of 3 or more defective components.

1. Check approximation conditions: n=1000 ≥ 50, p=0.002 < 0.1, so λ = np = 2
2. Rewrite the required probability: P(X ≥ 3) = 1 - P(X ≤ 2)
3. $$P(X ≤ 2) = e^{-2} \left( 1 + 2 + \frac{2^2}{2!} \right) = 5e^{-2} ≈ 0.6767$$
4. Final result: 1 - 0.6767 ≈ 0.323

## Common pitfalls

- **Wrong:** Calculating P(X ≥ 2) as 1 - P(X ≤ 2)
  - Why it fails: This is the most common off-by-one error for discrete cumulative probabilities, leading to an incorrect value that includes P(X=2) in the subtracted term
  - Correct: Rewrite all inequalities explicitly to confirm the upper limit of the cumulative probability you subtract: P(X ≥ k) = 1 - P(X ≤ k-1)
- **Wrong:** Assuming the variance of a Poisson distribution is different from its mean
  - Why it fails: Students often mix up Poisson properties with Binomial, where variance = np(1-p) < mean
  - Correct: Memorise that for any Poisson variable X, E(X) = Var(X) = λ, this is a unique identifying property
- **Wrong:** Applying Poisson approximation to Binomial when p > 0.1 or n < 20
  - Why it fails: The approximation loses significant accuracy outside the valid parameter range, leading to answers that are far from the exact Binomial value
  - Correct: Only use Poisson approximation if n ≥ 50 and np < 10, otherwise use the exact Binomial formula
- **Wrong:** Using the Poisson model for scenarios where events can occur in groups
  - Why it fails: This violates the 'events occur singly' condition, breaking the core assumptions of the distribution
  - Correct: Verify all 4 Poisson conditions before selecting the distribution for any problem
- **Wrong:** Rounding e^{-λ} to 2 or 3 decimal places during intermediate steps
  - Why it fails: CIE mark schemes deduct 1 accuracy mark for answers that are more than 0.01 off the correct value due to early rounding
  - Correct: Keep all intermediate values to at least 4 decimal places before giving your final answer to 3 significant figures

## Cheatsheet

| Scenario | Distribution | Mean | Variance |
| --- | --- | --- | --- |
| Independent events at constant rate over interval | Po(λ) | λ | λ |
| Binomial n large, p small, np < 10 | Po(np) | np | np |
| P(X ≤ k) | Cumulative table lookup | - | - |
| P(a ≤ X ≤ b) | P(X ≤ b) - P(X ≤ a-1) | - | - |

## What's next

Now that you have mastered the foundational properties and calculations for the Poisson distribution, you will be able to extend this knowledge to more advanced CIE 9709 topics including sums of independent Poisson variables, normal approximations for large λ values, and Poisson-based hypothesis testing. This is a high-weight exam topic that frequently appears combined with general probability rules, and accounts for up to 15% of the total marks on Paper 4. Mastering the core workflows here will help you avoid the most common lost-mark errors, and make subsequent statistical modelling topics far more intuitive.

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