Understanding the Poisson Distribution

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, assuming events happen at a constant average rate and independently of the time since the last event.

Common Applications

Predicting call center incoming calls per hour
Modeling website visitors per minute
Calculating mutation rates in genetics
Quality control (defects per manufactured unit)

Calculation Example

For λ=2.5 and k=3:

P(X=3) = (2.5³ × e⁻²·⁵) / 3! = (15.625 × 0.0821) / 6 ≈ 0.2138

Reference Table

λ Value	Typical Use Cases
λ < 1	Very rare events
1 ≤ λ ≤ 10	Common Poisson range
λ > 10	Approaches normal distribution

Frequently Asked Questions

What values can λ take?

λ must be a positive real number representing the average event rate per interval.

How large can k be?

The calculator handles k values up to 100 accurately. For k > 100, results use Stirling's approximation for factorial calculations.

When is Poisson distribution appropriate?

When events are independent, the average rate is constant, and two events cannot occur at exactly the same instant.

Poisson Distribution Calculator

Results

Probability Mass Function Visualization

Menu

Poisson Distribution Calculator - Rare Event Model

AI Assistant

Was this tool helpful?