The Poisson distribution

CIE A-Level Mathematics· 9709 Unit 4 Probability & Statistics 1/2· 45 min read

1. Core Definition and Valid Use Conditions★★☆☆☆⏱ 10 min

📘 Definition

Poisson Distribution

X ~ Po(λ)

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

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.

📐 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.

✓ Quick check

Confirm your understanding of Poisson conditions

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
Reveal answer
Number of heads in 100 coin flips —
This is a Binomial scenario with fixed number of trials, not events over an interval.

2. Poisson Probability Formula and Basic Calculations★★★☆☆⏱ 12 min

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
$P(X=3) = \frac{0.082085 \times 15.625}{6} ≈ 0.214$

3. Cumulative Poisson Probability Table Usage★★★☆☆⏱ 10 min

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.

📐 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

4. Poisson Approximation to the Binomial Distribution★★★★☆S1 / S2 only⏱ 13 min

🔬 Derivation

Goal:

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

Result:

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

5. Common Pitfalls

Wrong move:

Calculating P(X ≥ 2) as 1 - P(X ≤ 2)

Why:

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 move:

Rewrite all inequalities explicitly to confirm the upper limit of the cumulative probability you subtract: P(X ≥ k) = 1 - P(X ≤ k-1)

Wrong move:

Assuming the variance of a Poisson distribution is different from its mean

Why:

Students often mix up Poisson properties with Binomial, where variance = np(1-p) < mean

Correct move:

Memorise that for any Poisson variable X, E(X) = Var(X) = λ, this is a unique identifying property

Wrong move:

Applying Poisson approximation to Binomial when p > 0.1 or n < 20

Why:

The approximation loses significant accuracy outside the valid parameter range, leading to answers that are far from the exact Binomial value

Correct move:

Only use Poisson approximation if n ≥ 50 and np < 10, otherwise use the exact Binomial formula

Wrong move:

Using the Poisson model for scenarios where events can occur in groups

Why:

This violates the 'events occur singly' condition, breaking the core assumptions of the distribution

Correct move:

Verify all 4 Poisson conditions before selecting the distribution for any problem

Wrong move:

Rounding e^{-λ} to 2 or 3 decimal places during intermediate steps

Why:

CIE mark schemes deduct 1 accuracy mark for answers that are more than 0.01 off the correct value due to early rounding

Correct move:

Keep all intermediate values to at least 4 decimal places before giving your final answer to 3 significant figures

6. Quick Reference 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)

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 42
Poisson approximation to binomial
2022 · 41
Cumulative probability for defects
2021 · 43
Combined Poisson event counts

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.