# Definition:Exponential Distribution

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## Definition

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: $\var X = \beta^2$

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`