# Definition:Exponential Distribution

## 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

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