When do you use a Poisson distribution?

For the number of events in a fixed interval of time or space when the events occur independently, the mean number is proportional to the size of the interval, and two events practically never occur at the same instant.

What is special about the mean and variance of a Poisson distribution?

They are equal: both are λ. For a binomial distribution the variance is smaller than the mean, which helps you identify which model fits a set of data.

Is the Poisson table required in DSE M1?

No. The curriculum notes say using the Poisson distribution table is not required.

The Poisson Distribution (DSE M1)

A Poisson distribution counts events that occur independently at a constant average rate. If the mean number in the interval is λ, then X ~ Po(λ) with P(X = k) = e^−λ λᵏ / k! and E(X) = Var(X) = λ.

Solve The Poisson Distribution questions with AI

Po(λ) as the mean moves from 0.5 to 6. At λ = 2.5 the first three bars add up to 0.5438, the answer to Example 1. Try it yourself →

Poisson Distribution Formulas

ProbabilityP(X = k) = e^−λ λᵏ / k!, k = 0, 1, 2, …

Mean and varianceE(X) = λ, Var(X) = λ

Changing the intervalMean is proportional to the size of the interval: rate r per unit gives λ = rt for t units

No eventP(X = 0) = e^−λ

Worked Examples — The Poisson Distribution

Example 1: A cumulative Poisson probability

The number of calls received per minute follows a Poisson distribution with mean 2.5. Find the probability that at most 2 calls are received in a minute.

X ~ Po(2.5)
P(X ≤ 2) = e^−2.5(1 + 2.5 + 2.5²/2!)
= 6.625 e^−2.5
= 0.5438

Answer: 0.5438

Example 2: Rescaling the mean for a longer interval

Accidents occur at a junction at an average rate of 1.2 per week. Find the probability that exactly 4 accidents occur in 5 weeks.

Mean for 5 weeks: λ = 1.2 × 5 = 6, so Y ~ Po(6)
P(Y = 4) = e⁻⁶ × 6⁴ / 4!
= e⁻⁶ × 54 = 0.1339

Answer: 0.1339

💡 Exam Traps & Secrets

⚠️ λ must match the interval in the questionIf the rate is given per minute and the question asks about 3 minutes, use λ = 3 × rate. Using the per-minute mean for a 3-minute interval is the most common Poisson error.

⚠️ "At least one" is 1 − e^(−λ)A Poisson variable has no upper limit, so P(X ≥ 1) cannot be summed directly. Use the complement 1 − P(X = 0).

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Poisson probability)X ~ Po(3). Find P(X = 2) and P(X ≥ 2), and show the distribution as a bar chart.

(Rescaling λ)Emails arrive at an average of 4 per hour. Find the probability of receiving no email in 30 minutes.

(Finding λ)A Poisson variable has P(X = 0) = 0.2. Find its mean and P(X = 1).