The Poisson distribution models counts of rare, independent events over time or space.
How to use the Poisson Distribution Calculator
- Average rate (λ)
- Number of events (k)
Results update as you type, and every field accepts the units shown in its menu.
How it works
Raise λ to the power k, multiply by e^(−λ) and divide by k factorial.
Inputs explained
| Field | What to enter |
|---|---|
| Average rate (λ) | A number |
| Number of events (k) | A number |
What each result means
- P(X = k) = exp(-[Average rate (λ)])×[Average rate (λ)]^[Number of events (k)]÷fact([Number of events (k)])
- P(X = 0) = exp(-[Average rate (λ)])
- Standard deviation = sqrt([Average rate (λ)])
Example values
How the main result changes when Average rate changes and everything else stays at the example values above:
| Average rate (λ) | P(X = k) |
|---|---|
| 2 | 0.2707 |
| 3 | 0.224 |
| 4 | 0.1465 |
| 5 | 0.0842 |
| 6 | 0.0446 |
| 8 | 0.0107 |
Formula
P(X = k) = e^(−λ) λᵏ ÷ k!Worked example
Inputs — Average rate (λ): 4; Number of events (k): 2.
Results — P(X = k) = 0.146525; P(X = 0) = 0.018316; Standard deviation = 2.
Frequently asked questions
What is special about Poisson?
Its mean and variance are both λ.
When does it approximate the binomial?
When n is large and p is small, with λ = np.
What formula does the Poisson Distribution Calculator use?
P(X = k) = e^(−λ) λᵏ ÷ k!.