Definition:Exponential Distribution

Definition

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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.): exponential distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.): exponential distribution
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability Distributions