What is Poisson distribution?

Poisson distribution models the number of events occurring in a fixed interval, given a known average rate (λ). Examples: emails/hour, accidents/month, website visits/day.

What are the requirements?

Events occur independently, at a constant average rate, and in a fixed time/space interval. Events can't happen at exactly the same instant.

Lambda is the average rate of events. If you receive 5 emails per hour on average, λ = 5. It's both the mean AND variance of the distribution.

Binomial: fixed n trials, each with p probability. Poisson: events in continuous time/space at rate λ. Poisson approximates Binomial when n is large and p is small (np = λ).

What's the probability of no events?

P(X = 0) = e^(-λ). With λ = 3: P(0) ≈ 0.05 or 5%. Higher λ means lower chance of zero events.

Poisson Distribution Calculator

Calculate probabilities for rare events occurring at a constant rate.

Average Rate (λ)

Expected number of events

Number of Events (k)

Events to calculate probability for

Distribution

P(X = 3)

0 %

Exactly 3 events with λ = 5

P(X = 3)

P(X ≤ 3)

P(X ≥ 3)

Mean (μ)

= λ

Variance (σ²)

= λ (same as mean!)

Documentation

Poisson Distribution

The Poisson distribution models the probability of k events occurring in a fixed interval when events happen at a constant average rate λ.

PMF

Mean

Variance

Frequently Asked Questions

How to Use

About the Poisson

What it calculates

This poisson solves mathematical and statistical problems using standard formulas. Calculate probabilities for rare events (Poisson).

When to use it

Use for homework, data analysis, exam prep, or quick verification of manual calculations.

Example

Adjust the inputs above to model your specific scenario with the Poisson.

Explore More

Poisson Distribution Calculator

Distribution

0 %

Poisson Distribution

About the Poisson

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