# Definition:Exponential Distribution

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## Definition

This article needs to be linked to other articles.In particular: Link to what the two displayed lines actually meanYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.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 `{{MissingLinks}}` from the code. |

This article is complete as far as it goes, but it could do with expansion.In particular: The definition as given in both cited works is in a different format from this, notably such that the parameter $\lambda$ is $\dfrac 1 \beta$. Comes to the same thing in the end, but we need either to use a multidefinition format or to add an "also defined as". Which is standard? Source works need to be investigated. I may be due for another attack on Grimmett and Welsh.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.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 `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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

Then $X$ has the **exponential distribution with parameter $\beta$** if and only if:

- $\map X \Omega = \R_{\ge 0}$
- $\map \Pr {X < x} = 1 - e^{-\frac x \beta}$

where $0 < \beta$.

It is written:

- $X \sim \Exponential \beta$

## Also see

- Probability Density Function of Exponential Distribution
- Expectation of Exponential Distribution: $\expect X = \beta$
- Variance of Exponential Distribution: $\var X = \beta^2$

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

## Technical Note

The $\LaTeX$ code for \(\Exponential {\beta}\) is `\Exponential {\beta}`

.

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

`\Exponential \beta`

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**exponential distribution** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**exponential distribution** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions