A call centre receives an average of 4 calls per minute. What’s the probability it receives exactly 6 calls in the next minute? Or none at all? These questions can’t be answered with the binomial distribution covered elsewhere in this series, because there’s no fixed “number of trials” — calls can arrive at any moment, in any number, with no natural upper limit. This is precisely the situation the Poisson distribution was built to model.
What Is the Poisson Distribution?
The Poisson distribution describes the number of times an event occurs within a fixed interval of time or space, given that events happen at a known constant average rate and independently of the time since the last event. It’s named after French mathematician Siméon Denis Poisson, who developed it in 1837 while studying the number of wrongful convictions in French military courts.
Like the binomial distribution, Poisson is discrete — it counts whole numbers of occurrences (0, 1, 2, 3, and so on). But unlike binomial, there is no fixed number of “trials” and no concept of “failure” — you’re simply counting how many times something happens within a window, when in principle it could happen any number of times.
The Conditions for a Poisson Process
A situation is well-modelled by a Poisson distribution when four conditions roughly hold:
1. Events occur at a known, constant average rate over the interval in question, denoted λ (lambda). This rate can be per minute, per day, per square metre, per page — whatever unit of time or space the interval is measured in.
2. Events occur independently of one another — one arrival does not make the next arrival more or less likely.
3. Two events cannot occur at exactly the same instant — at a fine enough resolution, arrivals are separated in time.
4. The probability of an event occurring in a very short sub-interval is proportional to the length of that sub-interval, and the probability of more than one event in a very short sub-interval is negligible.
The Poisson Probability Formula
If X is the number of events occurring in a fixed interval with average rate λ, then X follows a Poisson distribution, written X ~ Poisson(λ). The probability of observing exactly k events is:
P(X = k) = (e−λ × λk) / k!
Here e is Euler’s number (≈ 2.71828), a mathematical constant that appears throughout probability and calculus wherever continuous growth or decay is involved. λ is the average rate of occurrence over the interval — notably, this single parameter defines the entire distribution, meaning the Poisson distribution’s mean and variance are both equal to λ, a distinctive and easily testable property.
Worked Example: Call Centre Arrivals
The call centre averages λ = 4 calls per minute. What is the probability of exactly 6 calls in the next minute?
P(X = 6) = (e−4 × 46) / 6!
Calculating each piece: e−4 ≈ 0.0183, 46 = 4,096, and 6! = 720.
P(X = 6) = (0.0183 × 4,096) / 720 ≈ 74.98 / 720 ≈ 0.1042
There’s roughly a 10.4% chance of exactly 6 calls in the next minute. Now consider a different, very practical question: what’s the probability of zero calls in a minute — useful for a manager trying to estimate idle time?
P(X = 0) = (e−4 × 40) / 0! = (0.0183 × 1) / 1 = 0.0183
Only about a 1.8% chance of a completely quiet minute — consistent with an average of 4 calls arriving steadily. Notice how the formula handles k = 0 cleanly: 4⁰ = 1 and 0! = 1 by definition, so P(X=0) reduces to simply e−λ.
Mean and Variance
The Poisson distribution has an unusually elegant property: its mean and variance are identical, both equal to λ.
Mean: μ = λ Variance: σ² = λ Standard deviation: σ = √λ
For the call centre example with λ = 4, the standard deviation is √4 = 2 calls per minute. This mean-equals-variance property is actually a useful diagnostic: if real count data has a variance noticeably larger than its mean (called overdispersion), the Poisson model may not be appropriate, and analysts often turn to a related distribution (the negative binomial) that allows variance to exceed the mean.
Worked Example: Rescaling the Rate
The call centre’s rate of 4 calls per minute translates to 240 calls per hour. What is the probability of receiving fewer than 2 calls in a 30-second window?
First, rescale λ to match the new interval: 4 calls per minute means 2 calls per 30 seconds, so λ = 2 for this window.
P(X < 2) = P(X = 0) + P(X = 1)
P(X = 0) = e−2 × 2⁰ / 0! = e−2 ≈ 0.1353
P(X = 1) = e−2 × 2¹ / 1! = 0.1353 × 2 = 0.2707
P(X < 2) = 0.1353 + 0.2707 ≈ 0.406
Rescaling λ to match whatever interval a question asks about — per second, per hour, per day — is the step students most often skip, and it changes every subsequent calculation if done incorrectly.
The Poisson Approximation to the Binomial
When a binomial distribution has a very large number of trials (n) and a very small probability of success (p), such that the mean np stays moderate, the Poisson distribution with λ = np gives an excellent and much simpler approximation. This is why Poisson historically arose from studying rare events — misprints on a page, deaths by horse kick in the Prussian cavalry, radioactive decay counts — situations where n is enormous (many opportunities for the event) but p is tiny (each individual opportunity rarely results in the event).
Real-World Applications
The Poisson distribution appears throughout fields that involve counting rare or randomly-timed events. In insurance and actuarial science, the number of insurance claims filed in a given month is often modelled as Poisson, directly informing premium pricing. In epidemiology, the number of disease cases in a region over a given period follows a Poisson model, used to detect unusual clusters that might signal an outbreak. In telecommunications and web infrastructure, the number of requests hitting a server in a given second is modelled as Poisson to size server capacity correctly — too little capacity causes outages during ordinary statistical fluctuation, not just unusual spikes. In quality control, the number of defects per unit area of manufactured material (like flaws per square metre of fabric) is a classic Poisson application.
Common Mistakes
Forgetting to rescale λ to match the interval in the question. As shown above, a rate given “per minute” must be converted before it can answer a question about “per 30 seconds” or “per hour” — using the wrong λ invalidates every subsequent calculation.
Applying Poisson when events aren’t independent. If one event genuinely makes a subsequent event more or less likely — for instance, a burst of related customer complaints after a single service outage — the independence assumption breaks down and a naive Poisson model will understate the true variability.
Confusing Poisson with binomial when a fixed n does exist. If a question specifies a fixed, finite number of trials with a per-trial probability (like “10 coin flips”), that’s binomial, not Poisson — Poisson is for open-ended counts over a continuous interval, not a pre-specified number of discrete attempts.
Practice Question
A website receives an average of 3 server errors per hour. What is the probability it experiences exactly 2 errors in the next hour?
Answer: λ = 3, k = 2.
P(X=2) = (e−3 × 3²) / 2! = (0.0498 × 9) / 2 = 0.4482 / 2 ≈ 0.224
There’s approximately a 22.4% chance of exactly 2 server errors in the next hour, given the historical average rate of 3 per hour.
References
1. Ross, S.M. (2020) A First Course in Probability. Pearson.
2. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
3. Wackerly, D., Mendenhall, W. and Scheaffer, R.L. (2014) Mathematical Statistics with Applications. Cengage.
4. Bortkiewicz, L. von (1898) Das Gesetz der kleinen Zahlen. Teubner.
5. Triola, M.F. (2022) Elementary Statistics. Pearson.