When the Poisson distribution applies
The Poisson distribution counts how many times an event happens in a fixed interval of time or space when events:
- occur independently of each other,
- happen one at a time,
- arrive at a constant average rate λ.
Typical examples: calls to a call center per hour, emails per minute, typos per page, goals per soccer match, mutations per DNA segment or defects per square meter of fabric.
The formulas
- P(X = k) = e^(−λ) × λ^k ÷ k!
- P(X ≤ k) = sum of P(X = i) for i from 0 to k
- Mean = variance = λ, standard deviation = √λ
Worked example
A help desk receives on average λ = 4 calls per hour. What is the chance of exactly 2 calls in the next hour?
- e^(−4) ≈ 0.018316.
- λ^k ÷ k! = 16 ÷ 2 = 8.
- P(X = 2) ≈ 0.018316 × 8 ≈ 14.65%.
- P(X ≤ 2) = e^(−4) × (1 + 4 + 8) ≈ 23.81%.
- P(X ≥ 3) = 1 − 23.81% ≈ 76.19%.
Probabilities for λ = 4
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 1.83% | 1.83% |
| 1 | 7.33% | 9.16% |
| 2 | 14.65% | 23.81% |
| 3 | 19.54% | 43.35% |
| 4 | 19.54% | 62.88% |
| 6 | 10.42% | 88.93% |
| 8 | 2.98% | 97.86% |
The most likely values sit around λ; when λ is a whole number, λ − 1 and λ are equally likely.
Changing the interval
λ scales with the interval. If a website gets 4 orders per hour on average, it gets λ = 2 per half hour and λ = 96 per day. Always match λ to the interval you ask about.
Poisson and binomial
When an event has a small probability p in each of many trials n, the number of successes is approximately Poisson with λ = n × p. For instance, if 1 in 1,000 items is defective, the number of defects in a batch of 2,000 is close to Poisson with λ = 2.
Checking the model
Compare the mean and the variance of your observed counts. If the variance is much larger, events cluster, for example rush hours or contagious effects, and a negative binomial model fits better. If it is much smaller, events are more regular than random.