Definition:Exponential Distribution

Definition

This article needs to be linked to other articles. In particular: Link to what the two displayed lines actually mean You 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.

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

• Expectation of Exponential Distribution: $\expect X = \beta$
• Variance of Exponential Distribution: $\var X = \beta^2$

This page has been identified as a candidate for refactoring of basic complexity. In particular: Move the following into its own page, along with a demonstration of its truth. See Probability Generating Function of Geometric Distribution for an example of how this is structured. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. 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 {{Refactor}} from the code.

The probability density function of $X$ is:

\(\ds \map {f_X} x\)

\(=\)

\(\ds \begin{cases} \dfrac 1 \beta e^{-\frac x \beta} & : x \ge 0 \\ 0 & : \text{otherwise} \end{cases}\)

• 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

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.