Statistics Reference
Poisson Distribution Explained
Give the Poisson distribution one number — the average count per window — and it hands back the probability of every possible count: no calls this hour, exactly three defects, ten or more arrivals. This guide works the formula on a concrete example, checks the assumptions honestly, and demonstrates numerically why the Poisson so often stands in for the binomial.
The Formula and Its One Parameter
P(X = k) = e^(−λ) λᵇ / k!
mean = λ variance = λ
λ (lambda) is the average number of events per window, estimated from history: 120 support calls over 30 hours → λ = 4 per hour. Everything else follows from that single number — including the distribution's spread, since the variance equals the mean. That identity doubles as a diagnostic: real count data whose variance far exceeds its mean is telling you the Poisson assumptions don't hold.
Worked Example: A Help Desk at λ = 4 Calls/Hour
- Exactly two calls: P(X = 2) = e⁻⁴ × 4² / 2! = 0.018316 × 16 / 2 ≈ 0.1465.
- A quiet hour (zero calls): P(X = 0) = e⁻⁴ ≈ 0.0183 — under 2% of hours.
- At most two calls: P(X ≤ 2) = e⁻⁴(1 + 4 + 8) ≈ 0.2381.
- Staffing question — more than two calls: 1 − 0.2381 = 0.7619: three-plus calls in about three hours of every four.
The Poisson distribution calculator reproduces each value from λ = 4 and k = 2, including the cumulative forms. Note how the window defines λ: the same desk over half-hours uses λ = 2, over days λ = 96 — state the window whenever you quote a rate.
The Assumptions Checklist
- One at a time. Events do not arrive in simultaneous batches (a bus of 40 tourists is one event, not 40).
- Independent. One event does not trigger or suppress another — contagion and cascades break the model.
- Steady rate. λ is constant across the window; if mornings and evenings differ, model them as separate windows with their own rates.
- No ceiling. The count could in principle be any size. If there is a hard maximum of n trials, that is the binomial's territory — the decision guide walks the choice.
The most common failure in practice is the third: rates that vary between windows inflate the variance above the mean (overdispersion), making Poisson tail probabilities too optimistic.
Poisson ≈ Binomial: A Numeric Demonstration
When n is large and p small (rule of thumb: n ≥ 100, np ≤ 10), the binomial's fixed ceiling stops mattering and only the rate λ = np survives. Concretely — 1,000 units, each defective with probability 0.003, so λ = 3:
| Count k | Exact binomial (n=1000, p=0.003) | Poisson (λ=3) |
|---|---|---|
| 0 | 0.049563 | 0.049787 |
| 2 | 0.224154 | 0.224042 |
Agreement to the third decimal, with the Poisson needing one parameter instead of two. This limit is why rare-event counts — defects, accidents, mutations — are modeled Poisson even when a finite n technically exists.
The Exponential Connection
The Poisson counts events per window; the exponential distribution times the gaps between them — two views of the same process sharing one λ. The link is visible in the formulas: P(zero events in a window of length t) = e^(−λt), which is exactly the exponential's probability that the first waiting time exceeds t. At the help desk, the 1.8% of quiet hours and the chance a gap between calls exceeds an hour are the same number.
Try the Poisson Distribution Calculator
Turn an average rate into exact and cumulative probabilities — every number in the worked example, automated.
Frequently Asked Questions
What is the Poisson distribution in simple terms?
It gives the probability of observing exactly k events in a fixed window of time or space when events occur independently at a steady average rate: calls per hour, defects per meter, typos per page. Its single parameter lambda is the average count per window, and the formula P(X = k) = e^(-lambda) lambda^k / k! turns that average into probabilities for every possible count.
What does lambda represent, and how do I get it?
Lambda is the expected number of events per window - usually estimated from history as total events divided by total windows (e.g. 120 calls over 30 hours gives lambda = 4 per hour). It scales with the window: if lambda is 4 per hour, it is 2 per half-hour and 96 per day. Lambda is simultaneously the distribution's mean and its variance.
What are the assumptions of the Poisson model?
Four checks: events occur one at a time (no simultaneous batches), independently (one arrival doesn't trigger another), at a constant average rate across the window you model, and with the count unbounded in principle. Rush hours violate the constant rate (model shorter windows), and contagious or clustered events violate independence.
Why does mean = variance matter, and what if my data's variance is larger?
Mean equals variance is the Poisson signature, so comparing them in real count data is a free diagnostic. Variance well above the mean (overdispersion) is common - it signals clustering or a rate that varies between windows - and makes Poisson probabilities overconfident. The negative binomial distribution is the standard remedy.
When does the Poisson approximate the binomial?
When n is large and p is small with np moderate - a common rule is n >= 100 and np <= 10. Set lambda = np. The approximation is excellent in that regime: for n = 1000, p = 0.003, the exact binomial P(X = 2) is 0.22415 versus Poisson 0.22404 - agreement to three decimals, with the Poisson needing only one parameter.
What is the difference between the Poisson and exponential distributions?
Two views of one process: the Poisson counts events per fixed window; the exponential measures the waiting time between consecutive events. If counts are Poisson with rate lambda, gaps are exponential with the same lambda, and P(zero events in a window) = e^(-lambda) is exactly the probability the first wait outlasts the window.