⏱️ Poisson Distribution Calculator

Calculate exact or cumulative probabilities for event counts occurring at an average rate.

📊 Statistics📐 Formula included🧠 Practice questions✓ Free

Average rate λ

Event count k

Probability

Please enter valid values.

Formula and Explanation

Calculate exact or cumulative probabilities for event counts occurring at an average rate.

Concept	Formula or rule
Poisson Distribution Calculator	P(X=k) = e⁻ˡᵃᵐᵇᵈᵃ λᵏ / k!

How to Use This Calculator

Enter the sample, counts, probabilities, or summary statistics requested above.
Keep paired lists in the same order and separate values with commas.
Select the correct test direction or sample/population option when available.
Press Calculate and interpret the result together with the supporting statistics.

Worked Example

Example: If a help desk averages 4 calls per minute, the probability of exactly 3 calls follows a Poisson model.

✓ P(X=3) = e⁻⁴×4³/3! ≈ 0.1954

Where This Is Used

School & coursework

Check homework, understand formulas, and verify step-by-step statistics exercises.

Research

Explore estimates, uncertainty, distributions, and relationships before using full statistical software.

Business & data analysis

Summarize variation, compare groups, and communicate data-driven findings clearly.

Common Mistakes to Avoid

⚠️

Using a mismatched time unit
Rate and observation interval must use the same unit.

⚠️

Ignoring changing rates
A basic Poisson model assumes a constant average rate.

⚠️

Using it for strongly dependent events
Events should occur independently enough for the model.

Practice Questions

1. If λ=2, what is the mean event count?

Answer: 2

2. What is the variance when λ=5?

Answer: 5

3. Can k be negative?

Answer: No.

Frequently Asked Questions

What is the Poisson mean?▾

λ.

What is the Poisson variance?▾

Also λ.

When can Poisson approximate binomial?▾

When n is large, p is small, and np is moderate.