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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions

1.3.6.6.12.

Double Exponential Distribution

Probability Density Function The general formula for the probability density function of the double exponential distribution is

\( f(x) = \frac{e^{-\left| \frac{x-\mu}{\beta} \right| }} {2\beta} \)

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.

plot of the double exponential probability density function

Cumulative Distribution Function The formula for the cumulative distribution function of the double exponential distribution is

\( F(x) = \begin{array}{ll} \frac{e^{x}} {2} & \mbox{for $x < 0$} \\ 1 - \frac{e^{-x}} {2} & \mbox{for $x \ge 0$} \end{array} \)

The following is the plot of the double exponential cumulative distribution function.

plot of the double exponential cumulative distribution function

Percent Point Function The formula for the percent point function of the double exponential distribution is

\( G(P) = \begin{array}{ll} \log(2p) & \mbox{for $p \le 0.5$} \\ -\log(2(1 - p)) & \mbox{for $p > 0.5$} \end{array} \)

The following is the plot of the double exponential percent point function.

plot of the double exponential percent point function

Hazard Function The formula for the hazard function of the double exponential distribution is

\( h(x) = \begin{array}{ll} \frac{e^{x}} {2 - e^{x}} & \mbox{for $x < 0$} \\ 1 & \mbox{for $x \ge 0$} \end{array} \)

The following is the plot of the double exponential hazard function.

plot of the double exponential hazard function

Cumulative Hazard Function The formula for the cumulative hazard function of the double exponential distribution is

\( H(x) = \begin{array}{ll} -log{(1 - \frac{e^{x}} {2})} & \mbox{for $x < 0$} \\ x + \log{(2)} & \mbox{for $x \ge 0$} \end{array} \)

The following is the plot of the double exponential cumulative hazard function.

plot of the double exponential cumulative hazard function

Survival Function The formula for the survival function of the double exponential distribution is

\( S(x) = \begin{array}{ll} 1 - \frac{e^{x}} {2} & \mbox{for $x < 0$} \\ \frac{e^{-x}} {2} & \mbox{for $x \ge 0$} \end{array} \)

The following is the plot of the double exponential survival function.

plot of the double exponential survival function

Inverse Survival Function The formula for the inverse survival function of the double exponential distribution is

\( Z(P) = \begin{array}{ll} \log(2(1-p)) & \mbox{for $p \le 0.5$} \\ -\log(2p) & \mbox{for $p > 0.5$} \end{array} \)

The following is the plot of the double exponential inverse survival function.

plot of the double exponential inverse survival function

Common Statistics
Mean μ
Median μ
Mode μ
Range \(-\infty \mbox{ to } \infty\)
Standard Deviation \( \sqrt{2}\beta \)
Skewness 0
Kurtosis 6
Coefficient of Variation \( \sqrt{2}(\frac{\beta} {\mu}) \)
Parameter Estimation The maximum likelihood estimators of the location and scale parameters of the double exponential distribution are

\( \hat{\mu} = \tilde{X} \)

\( \hat{\beta} = \frac{\sum_{i=1}^{N}|X_{i} - \tilde{X}|} {N} \)

where \(\tilde{X}\) is the sample median.

Software Some general purpose statistical software programs support at least some of the probability functions for the double exponential distribution.
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