# Poisson Distribution Calculator

Poisson distribution, a well-known discrete probability distribution, shows the likelihood of an event occurring within a specified time interval. These events are assumed to take place with a known average rate of success (given by the mean of the distribution) and independent of the last event.

If Random variable X, the number of events in a given interval, is distributed with mean (λ) and variance (σ² = λ), the probability of observing x events in n (many) trials when the probability of success λ /n is small, is obtained by:

P(X=x) = (e-λ * λx) / x!

, where x=0, 1, 2, 3, 4, … and P(X=x) is the Poisson probability and we say X follows a Poisson distribution with parameter λ.

The calculations provide either an exact probability or a cumulative probability, which refers to the probability that the Poisson random variable is greater or less than some specified limit. In contrast to a Binomial distribution that always has a finite upper bound, a Poisson random variable can take on any positive integer value. Poisson distribution has application in biological sciences, such as in predicting cell mutation within a large population.

If Random variable X, the number of events in a given interval, is distributed with mean (λ) and variance (σ² = λ), the probability of observing x events in n (many) trials when the probability of success λ /n is small, is obtained by:

P(X=x) = (e-λ * λx) / x!

, where x=0, 1, 2, 3, 4, … and P(X=x) is the Poisson probability and we say X follows a Poisson distribution with parameter λ.

The calculations provide either an exact probability or a cumulative probability, which refers to the probability that the Poisson random variable is greater or less than some specified limit. In contrast to a Binomial distribution that always has a finite upper bound, a Poisson random variable can take on any positive integer value. Poisson distribution has application in biological sciences, such as in predicting cell mutation within a large population.

## How to use this tool

1. Enter a non-negative numeric value for Mean (average rate of success) and Random Variable to compute individual and cumulative probabilities.

2. Verify your data is accurate in the table that appears.

3. Press the "Calculate Poisson Distribution" button to display results.

## Data Entry

Poisson random variable (x) | |

Average rate of success (λ) |

## Feedback

Have a question or a feature request about this tool? Feel free to reach out to us and let us know! We're always looking for ways to improve!## References

### This online tool may be cited as follows

## MLA | "Quest Graph™ Poisson Distribution Calculator." AAT Bioquest, Inc., 1 Jun. 2023, https://www.aatbio.com/tools/poisson-distribution-online-calculator. |

## APA | AAT Bioquest, Inc. (2023, June 1). Quest Graph™ Poisson Distribution Calculator. AAT Bioquest. https://www.aatbio.com/tools/poisson-distribution-online-calculator. |