The Poisson distribution is the right tool for counts of events that happen at a steady rate over time or space, where each individual event is rare but the count adds up. Customers arriving at a café in an hour, defects per yard of fabric, typos per page — these are Poisson, when the conditions hold. Here is what the conditions are, the formula, and a full numerical example.
When the Poisson Is the Right Model
A random variable X follows a Poisson distribution when it counts the number of events that occur in a fixed interval (time, length, area, volume), and three conditions hold:
- Events occur independently of one another.
- Events occur at a constant average rate over the interval.
- No two events occur at exactly the same instant — events are isolated, not clumped together.
The single parameter is λ (lambda), the expected number of events in the chosen interval. λ has to be positive but does not have to be a whole number.
Notice what the Poisson is not. It is not a model for the probability of a single event. It is not for outcomes with a hard upper limit (binomial is the right tool there). And it is not for events that come in clumps — if customers arrive in groups, the Poisson model breaks.
The Poisson Formula
If X ~ Poisson(λ), then
P(X = k) = (λ^k · e^(−λ)) / k!
for k = 0, 1, 2, 3, ... (every nonnegative integer is possible, even when λ is small — it just becomes very unlikely for large k).
Two features of the Poisson are unusual and worth remembering:
- E(X) = λ and Var(X) = λ. The mean and variance are equal. This is a strong empirical check: if your sample mean and sample variance are wildly different, your data is probably not Poisson.
- The distribution has no upper bound. There is a tiny but nonzero probability of seeing 100 events in an interval where you expect 3.
The Worked Example
A small coffee shop receives an average of 4 customers per 15 minutes during the mid-afternoon. Let X = number of customers in a randomly chosen 15-minute window. The conditions hold (independent arrivals, steady rate, isolated events), so X ~ Poisson(4).
Step 1 — Mean and variance. E(X) = λ = 4. Var(X) = λ = 4, so σ = 2.
Step 2 — P(X = 5): exactly 5 customers.
P(X = 5) = (4^5 · e^(−4)) / 5! = (1024 · 0.01832) / 120 ≈ 18.76 / 120 ≈ 0.1563
So about 15.6% of 15-minute windows see exactly 5 customers.
Step 3 — P(X = 0): no customers.
P(X = 0) = (4^0 · e^(−4)) / 0! = (1 · 0.01832) / 1 ≈ 0.0183
A quiet window with zero arrivals happens about 1.8% of the time.
Step 4 — P(X ≥ 6): six or more customers.
Use the complement to avoid an infinite sum:
P(X ≥ 6) = 1 − P(X ≤ 5) = 1 − Σ from k = 0 to 5 of P(X = k)
The individual probabilities, computed the same way:
- P(0) ≈ 0.0183
- P(1) ≈ 0.0733
- P(2) ≈ 0.1465
- P(3) ≈ 0.1954
- P(4) ≈ 0.1954
- P(5) ≈ 0.1563
Sum ≈ 0.7852. So P(X ≥ 6) ≈ 1 − 0.7852 ≈ 0.2148.
About 21% of 15-minute windows see six or more customers. Note that P(3) and P(4) are equal — Poisson probabilities peak at k = λ when λ is an integer, with the modes at both λ − 1 and λ.
Rescaling: Changing the Interval
The Poisson scales linearly with the interval. If λ = 4 per 15 minutes, then over a full hour the average is 4 × 4 = 16 customers, and the relevant distribution is Poisson(16). The same formula applies; you just use the rate that matches the interval you care about.
This is why the parameter is often described as a rate × interval product. A rate of 0.5 events per minute over 10 minutes gives λ = 5. Over 20 minutes, λ = 10. Always match the rate's units to the interval before plugging in.
Poisson as the Limit of Binomial
When n is large and p is small while np is moderate, the binomial distribution is very close to a Poisson with λ = np. The classic textbook scenario: a manufacturing line produces 10,000 parts a day with a 0.0003 defect rate. The number of defects is technically Binomial(10000, 0.0003), but C(10000, k) gets unwieldy fast. The Poisson approximation uses λ = 10000 × 0.0003 = 3, and gives essentially the same probabilities with much easier arithmetic.
A common rule of thumb: use the Poisson approximation when n ≥ 100 and p ≤ 0.01 (so np stays modest). The bigger n and the smaller p, the better the approximation.
Common Mistakes
The first is using the wrong λ. A problem might give "0.5 events per hour" and ask about a 3-hour window. λ for that window is 0.5 × 3 = 1.5, not 0.5. Always check the interval.
The second is forgetting that Poisson has no upper bound. P(X ≥ k) is a tail probability that goes all the way to infinity, but practically you only need to sum until the terms become negligible. Or use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1).
The third is assuming Poisson when the events are not independent or arrive in clumps. If two customers tend to arrive together, the independence assumption is broken. The variance of the count will exceed the mean (overdispersion), and a different model — often the negative binomial — fits better.
Getting Help
The Poisson is the natural counterpart to the binomial for rare events; if you have not seen binomial yet, the binomial distribution walkthrough covers the four conditions and the formula. For computing expected values like λ itself in the discrete setting, expected value and variance shows the general machinery the Poisson plugs into.
Conclusion
The Poisson distribution models counts of events over a fixed interval when the events are independent and occur at a steady rate. Its single parameter λ is both the mean and the variance, and the formula P(X = k) = (λ^k · e^(−λ)) / k! gives the probability of any specific count. For "at least k" probabilities, use the complement. For very large n and very small p, the Poisson approximates the binomial with λ = np — the easier formula for the same answer. Match λ to the interval you actually care about, and the rest is plugging in.
