Hong Life CalculatorsUnderstanding the Poisson distribution calculator — concept and principle
The Poisson distribution gives the probability of seeing exactly k events in a fixed interval of time or space when events happen independently at a constant average rate λ. Classic examples: calls arriving at a help desk per hour, typos per page, emergency admissions per night, defects per square metre of material, or radioactive decays per second. The calculator returns P(X = k), the cumulative P(X ≤ k) and the tail P(X > k).
How it is calculated
- P(X = k) = λᵏ·e⁻λ ÷ k!, for k = 0, 1, 2, …
- P(X ≤ k) = sum of P(X = i) for i = 0 to k.
- P(X > k) = 1 − P(X ≤ k).
λ is the expected number of events in the interval you care about, so it must match that interval: a rate of 6 per hour means λ = 3 for a 30-minute window. The mean and the variance of a Poisson variable are both equal to λ.
The principle
Split the interval into n tiny pieces, each with a small chance p = λ/n of one event and practically no chance of two. The count is then binomial(n, λ/n). As n grows, C(n, k)(λ/n)ᵏ approaches λᵏ/k! and (1 − λ/n)ⁿ⁻ᵏ approaches e⁻λ, which gives the Poisson formula. That is why it is called the law of rare events: many opportunities, each unlikely, at a steady rate. The model assumes independence and a constant rate within the interval.
Worked example
- Defaults: λ = 3, k = 2. Note e⁻³ ≈ 0.0497871.
- P(X = 0) = e⁻³ ≈ 0.0497871; P(X = 1) = 3e⁻³ ≈ 0.1493612.
- P(X = 2) = 3²·e⁻³ ÷ 2! = 4.5e⁻³ ≈ 0.2240418077.
- P(X ≤ 2) = (1 + 3 + 4.5)·e⁻³ = 8.5e⁻³ ≈ 0.4231900811.
- P(X > 2) = 1 − 0.4231900811 = 0.5768099189.
| Mean rate λ | P(X = k) |
|---|---|
| 0.75 | 0.132853093 |
| 1.5 | 0.2510214302 |
| 2.25 | 0.2667917872 |
| 3 | 0.2240418077 |
| 3.75 | 0.1653591506 |
| 4.5 | 0.1124785899 |
| 6 | 0.0446175392 |
How to read the result
With an average of 3 events, getting exactly 2 happens about 22% of the time, and more than 2 happens in about 58% of intervals. Staffing and capacity decisions often use the upper tail: find the smallest k where P(X > k) falls below an acceptable risk. The standard deviation is √λ, so for λ = 3 typical counts range roughly from 0 to 6.
Common mistakes
- Using a λ for the wrong interval length.
- Applying the model when events cluster (bursts of calls, outbreaks); then the variance exceeds the mean and a negative binomial model may fit better.
- Reading P(X ≤ k) when the question asks "more than k".
Frequently asked questions
Can λ be a decimal?
Yes. λ is an average and can be any positive number, such as 0.7 events per day. k must be a whole number.
When does the Poisson approximate the binomial well?
When n is large and p is small, with λ = np; a common rule of thumb is n ≥ 20 and p ≤ 0.05.
What is the probability of zero events?
P(X = 0) = e⁻λ, about 5% for λ = 3.
Sources: Sheldon M. Ross, A First Course in Probability, Pearson — Poisson random variables · NIST/SEMATECH e-Handbook of Statistical Methods, Poisson Distribution