# Poisson Distribution Calculator

Calculate exact, cumulative, and upper-tail Poisson probabilities from an expected event rate

> Canonical page: https://elysiatools.com/en/tools/poisson-distribution-calculator

- **Category:** Math & Numbers

- **Keywords:** poisson distribution, event rate, lambda, probability, statistics

## Overview

Calculate Poisson distribution probabilities.

Use this for independent event counts over a fixed interval when the average rate is known. It returns P(X = k), P(X <= k), P(X >= k), mean, variance, and standard deviation.

## Inputs

- **Rate Lambda** (number)
- **Occurrences** (number)
- **Probability Mode** (select)
- **Decimal Places** (number)

## When to use

- When predicting the number of customer arrivals or service requests within a specific timeframe.
- When estimating the probability of a certain number of defects or failures in a batch of manufactured products.
- When analyzing network traffic to determine the likelihood of server requests exceeding capacity in a given minute.

## How it works

- Enter the expected average rate of events (Rate Lambda) for your given time or space interval.
- Specify the target number of occurrences (k) you want to calculate the probability for.
- Select the probability mode to focus on exact, at most, or at least k occurrences, and adjust the decimal precision.
- The calculator applies the Poisson probability mass function to return the exact, cumulative, and upper-tail probabilities in JSON format.

## Use cases

- Call center staffing optimization based on expected hourly call volumes.
- Manufacturing quality assurance to predict the frequency of rare defects per production run.
- Website server load management by estimating the probability of traffic spikes per minute.

## Frequently asked questions

### What is Lambda in a Poisson distribution?

Lambda (λ) represents the average number of times an event occurs within a fixed interval of time or space. It is the primary parameter needed for Poisson calculations.

### What is the difference between exact, at most, and at least probabilities?

Exact calculates the probability of exactly k events occurring. 'At most' calculates the cumulative probability of k or fewer events. 'At least' calculates the probability of k or more events.

### Can Lambda be a decimal number?

Yes, the average rate (Lambda) can be a decimal or fractional value, even though the target number of actual occurrences (k) must be a whole number.

### When should I use the Poisson distribution instead of the binomial distribution?

Use the Poisson distribution when you are counting the number of events over a continuous interval, the total number of possible trials is very large or unknown, and the probability of a single event is small.

### What are the mean and variance of a Poisson distribution?

In a Poisson distribution, both the mean and the variance are exactly equal to the expected rate, Lambda (λ).

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