# Definition:Poisson Distribution

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## 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): Entry:**Poisson distribution** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Poisson distribution** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Poisson distribution**

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