1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions
1.3.6.6.12. |
Double Exponential Distribution |
\( f(x) = \frac{e^{-\left| \frac{x-\mu}{\beta} \right| }} {2\beta} \)
where μ is the location parameter and β is the scale parameter. The case where μ = 0 and β = 1 is called the standard double exponential distribution. The equation for the standard double exponential distribution is
\( f(x) = \frac{e^{-|x|}} {2} \)
Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.
Note that the double exponential distribution is also commonly referred to as the Laplace distribution.
The following is the plot of the double exponential probability density function.
\( F(x) = \begin{array}{ll} \frac{e^{x}} {2} & \mbox{for $x < 0$} \\ 1 - \frac{e^{-x}} {2} & \mbox{for $x \ge 0$} \end{array} \)
The following is the plot of the double exponential cumulative distribution function.
\( G(P) = \begin{array}{ll} \log(2p) & \mbox{for $p \le 0.5$} \\ -\log(2(1 - p)) & \mbox{for $p > 0.5$} \end{array} \)
The following is the plot of the double exponential percent point function.
\( h(x) = \begin{array}{ll} \frac{e^{x}} {2 - e^{x}} & \mbox{for $x < 0$} \\ 1 & \mbox{for $x \ge 0$} \end{array} \)
The following is the plot of the double exponential hazard function.
\( H(x) = \begin{array}{ll} -log{(1 - \frac{e^{x}} {2})} & \mbox{for $x < 0$} \\ x + \log{(2)} & \mbox{for $x \ge 0$} \end{array} \)
The following is the plot of the double exponential cumulative hazard function.
\( S(x) = \begin{array}{ll} 1 - \frac{e^{x}} {2} & \mbox{for $x < 0$} \\ \frac{e^{-x}} {2} & \mbox{for $x \ge 0$} \end{array} \)
The following is the plot of the double exponential survival function.
\( Z(P) = \begin{array}{ll} \log(2(1-p)) & \mbox{for $p \le 0.5$} \\ -\log(2p) & \mbox{for $p > 0.5$} \end{array} \)
The following is the plot of the double exponential inverse survival function.
| Mean | μ |
| Median | μ |
| Mode | μ |
| Range | \(-\infty \mbox{ to } \infty\) |
| Standard Deviation | \( \sqrt{2}\beta \) |
| Skewness | 0 |
| Kurtosis | 6 |
| Coefficient of Variation | \( \sqrt{2}(\frac{\beta} {\mu}) \) |
\( \hat{\mu} = \tilde{X} \)
\( \hat{\beta} = \frac{\sum_{i=1}^{N}|X_{i} - \tilde{X}|} {N} \)
where \(\tilde{X}\) is the sample median.
