# Definition:Exponential Distribution

## Definition

Let $X$ be a continuous random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

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

$X \left({\Omega}\right) = \R_{\ge 0}$
$\Pr \left({X < x}\right) = 1 - e^{-\frac x \beta}$

where $0 < \beta$.

It is written:

$X \sim \operatorname{Exp} \left({\beta}\right)$

The probability density function of $X$ is:

 $\displaystyle f_X \left({x}\right)$ $=$ $\displaystyle \begin{cases} \frac 1 \beta e^{-\frac x \beta} & : x \ge 0 \\ 0 & : \text{otherwise} \end{cases}$ $\quad$ $\quad$

## Also see

• Results about the exponential distribution can be found here.