# Definition:Poisson Distribution

Jump to navigation
Jump to search

## Definition

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ has the **Poisson distribution with parameter $\lambda$** (where $\lambda > 0$) if and only if:

- $\Img X = \set {0, 1, 2, \ldots} = \N$

- $\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$

It is written:

- $X \sim \Poisson \lambda$

## Also denoted as

Some sources denote this as:

- $X \sim \map {\operatorname {Pois} } \lambda$

Some sources use $\mu$ instead of $\lambda$, but this can cause confusion with instances where $\mu$ is used for the expectation.

However, as the expectation of $\Poisson \lambda$ is also $\lambda$, this may not be as much of a confusion as all that.

## Also see

- Poisson Distribution Gives Rise to Probability Mass Function satisfying $\map \Pr \Omega = 1$.

- Results about
**the Poisson distribution**can be found**here**.

## Source of Name

This entry was named for Siméon-Denis Poisson.

## Technical Note

The $\LaTeX$ code for \(\Poisson {\lambda}\) is `\Poisson {\lambda}`

.

When the argument is a single character, it is usual to omit the braces:

`\Poisson \lambda`

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.2$: Examples: $(8)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Poisson distribution** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Poisson distribution** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Poisson distribution** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Appendix $13$: Probability Distributions

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: This link should be for Cumulative Distribution Function of Poisson DistributionIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 39$: Probability Distributions: Poisson Distribution: $39.2$