# Poisson Distribution Calculator

Free Poisson distribution calculator. Calculate the probability of k events occurring given an average rate λ.

## What this calculates

Calculate Poisson probabilities for events that occur at a known average rate. Enter the average rate (lambda) and desired number of events to find exact and cumulative probabilities.

## Inputs

- **Average Rate (λ)** — min 0.001, max 500 — The average number of events per interval (must be positive).
- **Number of Events (k)** — min 0, max 170 — The number of events to calculate the probability for.

## Outputs

- **P(X = k)** — Probability of exactly k events.
- **P(X ≤ k)** — Probability of k or fewer events.
- **P(X ≥ k)** — Probability of k or more events.
- **Expected Value E(X)** — The expected number of events (equals λ).
- **Standard Deviation** — Standard deviation of the distribution (√λ).
- **Formula** — formatted as text — The Poisson probability formula used.

## Details

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

PMF:
P(X = k) = (λ^k × e^(-λ)) / k!

Key Properties

- Mean = λ

- Variance = λ

- Standard deviation = √λ

Common Applications

- Calls to a call center per hour

- Website visits per minute

- Defects per unit of product

- Accidents per month

## Frequently Asked Questions

**Q: What are the conditions for using a Poisson distribution?**

A: The Poisson distribution applies when: (1) events occur independently, (2) the average rate λ is constant over the interval, (3) two events cannot occur at exactly the same instant, and (4) you are counting the number of events in a fixed interval of time or space.

**Q: How is Poisson related to the binomial distribution?**

A: The Poisson distribution is a limiting case of the binomial distribution. When n is large and p is small (but n × p = λ is moderate), the binomial distribution B(n, p) approaches the Poisson distribution with parameter λ = n × p. A rule of thumb: use Poisson when n ≥ 20 and p ≤ 0.05.

**Q: Why are the mean and variance equal in a Poisson distribution?**

A: This is a unique property of the Poisson distribution. Both the mean and variance equal λ. If you observe that the sample variance is much larger than the sample mean, the data may be overdispersed and a negative binomial distribution might be more appropriate.

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Source: https://vastcalc.com/calculators/statistics/poisson-distribution
Category: Statistics
Last updated: 2026-04-21