What Is Poisson Distribution?
The Poisson distribution models the number of events that occur in a fixed interval of time or space when events happen independently at a constant average rate λ. The probability of exactly k events is P(X = k) = e^(−λ) λ^k ÷ k!.
Enter λ and k, choose whether you want exactly k, at most, at least, fewer than, more than or a range, and the calculator returns the probability with the mean, variance, standard deviation, mode and the chance of no events. A table lists P(X = k) and the cumulative values.
The article below gives the formulas, variables and a worked help-desk example, and answers common questions about Poisson versus binomial.
How to Use the Poisson Distribution Calculator
- Enter the average number of events per interval, λ.
- Choose exactly, at most, at least or a range.
- Enter k (and k₂ for a range).
- Read the probability and the charts.
Poisson Distribution Formula
Probability of k events
Cumulative probability
Mean and variance
Poisson Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| λ | Average number of events per interval | mean and variance |
| k | Number of events observed | 0, 1, 2, … |
| P(X = k) | Probability of exactly k events | |
| e | Euler's number | ≈ 2.71828 |
Poisson Distribution Worked Example
A help desk receives 4 calls per hour on average. The chance of exactly 6 calls in an hour is e^(−4) × 4⁶ ÷ 6! = 0.018316 × 4096 ÷ 720 = 0.1042, about 10.4%. The chance of at most 2 calls is 0.0183 + 0.0733 + 0.1465 = 0.2381.
With the example values loaded in the calculator above, the steps are:
- P(X = k) = e^(−λ) λ^k ÷ k! = e^(−4) × 4^6 ÷ 6! = 0.10419563.
- Cumulative P(X ≤ 6) = Σ from i = 0 to 6 of P(X = i) = 0.88932602.
- Mean and variance both equal λ = 4.
Result: P(X = 6) = 0.10419563; Mean = variance = λ = 4.
How to Read a Poisson Probability
The Poisson distribution gives the chance of k events in a fixed interval when events occur independently at a steady average rate λ. With λ = 4 calls per hour, the chance of exactly 6 calls is about 0.1042, or 10.4%.
The mean and variance are both λ, so the standard deviation is √λ = 2. The most likely count is the whole number just below λ, and for λ = 4 the counts 3 and 4 are equally likely. The table on this page lists the probability of each count.
Where Poisson Distribution Is Used
Arrivals and calls
Model calls, customers or emails per hour when they arrive independently at a steady average rate.
Rare events
Count accidents, defects or typos per unit area or time.
At least one
Find the chance of at least one event, which is 1 − e^(−λ).
Comparing with the binomial
Check whether a Poisson approximation is close to a binomial with large n and small p.
Common Poisson Distribution Mistakes to Avoid
- The interval for λ must match the interval for the count. If λ is per hour, the count is per hour.
- If the variance of your data is much larger than the mean, a Poisson model is too tight.
- Events must be independent. Events that cluster, such as aftershocks, break this assumption.
- Scale λ with the interval. 4 calls per hour is 2 per half hour.
Poisson Distribution Terms and Definitions
- λ (lambda)
- The average number of events per interval.
- Overdispersion
- Variance larger than the mean, which a Poisson model cannot capture.
Poisson Distribution FAQs
When should I use the Poisson distribution?
Use it to count events that occur independently at a constant average rate in a fixed interval of time or space, such as calls per hour or flaws per metre of cable.
How do I find the probability of at least one event?
Use 1 − P(0) = 1 − e^(−λ). With λ = 4, P(0) = e^(−4) ≈ 0.0183, so the chance of at least one is about 0.9817.
What is the difference between Poisson and binomial?
A binomial has a fixed number of trials n, each with probability p. A Poisson has no fixed n and counts events at an average rate. For large n and small p, a Poisson with λ = np approximates the binomial.
What are the mean and variance of a Poisson distribution?
Both equal λ. A Poisson with λ = 4 has mean 4, variance 4 and standard deviation 2.