# 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

- Expectation of Exponential Distribution: $E \left({X}\right) = \beta$
- Variance of Exponential Distribution: $\operatorname {var} \left({X}\right) = \beta^2$

- Results about
**the exponential distribution**can be found here.