Maths › Further Statistics 1 › The Poisson distribution
The Poisson distribution
Count events that arrive at random in a fixed window and one number describes everything. That number is 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.
- State the conditions a Poisson model requires and check them against a context.
- 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.
COMMON MISCONCEPTION
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 and no n and p to find: the mean rate λ over the window is the only parameter, and it doubles as the variance.
The booklet's table of standard discrete distributions gives the Poisson row in full, probability function, mean and variance together. Scaling the window scales λ with it. If calls arrive at Po(3) per minute, then over five minutes they arrive at Po(15). The same additivity lets independent Poisson counts be pooled, so Po(λ) plus 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), so P(X = 2) = e⁻³ × 3²/2! = 0.224.
P(X ≤ 1) = e⁻³(1 + 3) = 0.199.
The mean and the variance are both 3. That equality is the model's signature and the first thing to test against data.
When the model is allowed
Three conditions have to hold before you may write Po(λ). Events must occur singly, so no two arrive at exactly the same instant. They must occur independently, so one event neither invites nor blocks another. And they must occur at a constant average rate across the window.
Real contexts break these all the time. Goals in football cluster because a team that scores changes how both sides play. Calls to a helpline peak at nine in the morning, so the rate is not constant over a day. Examiners like a sentence naming the condition you doubt and saying why the context threatens it.
The binomial has its own list, worth holding beside it: a fixed number of trials, two outcomes per trial, independent trials, and a constant probability of success. Its mean is np and its variance np(1 − p), so the variance always falls short of the mean. A data set with variance above its mean rules out the binomial immediately, and one with variance well below the mean casts doubt on Poisson.
WORKED EXAMPLE
Is the model reasonable?
A garage records the number of customers arriving each hour between 8am and 6pm. Over 200 hours the mean is 6.1 and the variance 9.8. Comment on a Poisson model.
Poisson demands equal mean and variance. Here the variance is roughly 1.6 times the mean, so the counts vary more than the model allows.
The likely culprit is the constant rate condition. Lunchtime and the end of the working day are busier than mid-morning, so pooling all ten hours mixes several rates together.
A Poisson model fitted hour by hour, or restricted to a quiet band of the day, would stand a far better chance.
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 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.
GUIDED PRACTICE
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, giving Po(11).
The variance equals the mean, so it is 11. Both the pooling and the stretching happen on λ alone.
ASSESSMENT FOCUS
- Write 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 whenever suitability is raised.
- Name a specific condition when asked whether the model fits. 'Events occur at a constant rate' scores where 'the conditions hold' does not.
- For the binomial approximation, say that n is large and p is small, then set λ = np.
- The additive property needs independence. Say so before you pool two counts.
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
Over four weeks the count is Po(8), so both the mean and the variance are 8.
Poisson counts events occurring singly, independently and at a constant average rate, with E(X) = Var(X) = λ.
Independent Poissons add and windows scale λ, and 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.
Or read them with their worked answers on the the poisson distribution questions page.
CHECK YOUR PROGRESS
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- Model random events with Po(λ) and compute probabilities on a calculator.
- State the conditions a Poisson model requires and check them against a context.
- 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.