Definition:Exponential Distribution
Definition
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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:
- $X \sim \operatorname{Exp} \left({\beta}\right)$
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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.
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