Definition:Exponential Distribution
Definition
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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: $\var X = \beta^2$
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The probability density function of $X$ is:
| \(\displaystyle \map {f_X} x\) | \(=\) | \(\displaystyle \begin{cases} \dfrac 1 \beta e^{-\frac x \beta} & : x \ge 0 \\ 0 & : \text{otherwise} \end{cases}\) | $\quad$ | $\quad$ |
- 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
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