Maths › Further Statistics 1 › The Poisson distribution
The Poisson distribution
Count events that arrive at random in a fixed window and one parameter describes everything: the mean rate. It also happens to be the variance, which makes the model easy to test and easy to break.
Builds on The binomial distribution and Discrete random variables and expectation.
IN THIS TOPIC
- Model random events with Po(λ) and compute probabilities on a calculator.
- Use E(X) = Var(X) = λ, and the additive property for independent counts.
- Approximate B(n, p) by Po(np) when n is large and p is small.
WHAT YOU PROBABLY THINK
The Poisson distribution needs both a number of trials and a probability, just as the binomial does.
One parameter, counted over a window
The Poisson distribution counts events landing at random in a fixed interval of time or space, with no upper limit on the count. There are no trials to count, so the opening claim is looking for machinery that does not exist: the mean rate λ over the window is the only parameter, and it doubles as the variance.
Scaling the window scales λ with it: if calls arrive at Po(3) per minute, then in five minutes they arrive at Po(15). The same additivity lets independent Poisson counts be pooled: Po(λ) + Po(μ) is Po(λ + μ).
WORKED EXAMPLE
Probabilities from a rate
Faults occur in a cable at a mean rate of 3 per kilometre, at random. Find the probability of exactly 2 faults in a kilometre, and of at most 1.
X ~ Po(3): P(X = 2) = e⁻³ × 3²/2! = 0.224.
P(X ≤ 1) = e⁻³(1 + 3) = 0.199.
The mean and variance are both 3, which is the model's signature and the first thing to test against data.
Standing in for the binomial
When n is large and p is small, B(n, p) is awkward to compute and nearly Poisson in shape. Taking λ = np gives an approximation that improves as p shrinks, because the binomial's variance np(1 − p) closes on np exactly when 1 − p is close to 1.
WORKED EXAMPLE
Rare events in many trials
A machine produces items with a 2% fault rate, independently. For a batch of 100, approximate the probability of no faults, and compare with the exact binomial value.
λ = np = 2, so P(X = 0) ≈ e⁻² = 0.135.
Exactly: 0.98¹⁰⁰ = 0.133. The two agree to two decimal places, and the gap narrows further for smaller p.
YOUR TURN
Pooling two rates
Emails arrive at Po(4) per hour and texts at Po(1.5) per hour, independently. Find the distribution of the total number of messages in a two-hour period, and its variance.
Show the working
Per hour the total is Po(4 + 1.5) = Po(5.5) by the additive property.
Over two hours the window doubles: Po(11).
The variance equals the mean, so it is 11. Both the pooling and the stretching happen on λ alone.
THE EXAM BIT
- State the distribution as X ~ Po(λ) with λ evaluated for the window in the question.
- Scale λ to the interval asked about before computing anything; per-minute rates rarely match per-question windows.
- Mean equal to variance is the Poisson's defining check; quote it when asked whether the model is suitable.
- For the binomial approximation, say that n is large and p is small, then set λ = np.
CHECK YOURSELF
Accidents at a junction follow Po(2) per week. Find the mean and variance of the number of accidents in a four-week month.
Show a hint
Scale λ with the window.
Show the answer
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Poisson counts random events in a window: one parameter λ, with E(X) = Var(X) = λ.
Independent Poissons add, and windows scale λ; B(n, p) becomes Po(np) for large n and small p.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Model random events with Po(λ) and compute probabilities on a calculator.
- Use E(X) = Var(X) = λ, and the additive property for independent counts.
- Approximate B(n, p) by Po(np) when n is large and p is small.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.