1.3.6.6.12. Double Exponential Distribution

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) = EXP(-|(x-mu)/beta|)/(2*beta)

where is the location parameter and beta is the scale parameter. The case where = 0 and beta = 1 is called the standard double exponential distribution. The equation for the standard double exponential distribution is

f(x) = EXP(-|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.

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) = EXP(x)/2 for x < 0,
F(x) = 1 - EXP(-x)/2 for x >= 0

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) = LOG(2*p) for p <= 0.5,
G(p) = -LOG(2*(1-p)) for p > 0.5

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) = EXP(x)/(2-EXP(x)) for x < 0, h(x) = 1 for x >= 0

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) = -LOG(1 - EXP(x)/2) for x < 0, H(x) = x + LOG(2) for x >= 0

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

plot of the double exponential cumulative hazard function

Survival Function

The double exponential survival function can be computed from the cumulative distribution function of the double exponential distribution.

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) = LOG(2*(1-p)) for p <= 0.5; Z(p) = -LOG(2*p) for p > 0.5

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

plot of the double exponential inverse survival function

Common Statistics

Mean
Median
Mode
Range	Negative infinity to positive infinity
Standard Deviation
Skewness	0
Kurtosis	6
Coefficient of Variation

Parameter Estimation

The maximum likelihood estimators of the location and scale parameters of the double exponential distribution are

muhat = XMEDIAN

betahat = SUM[|X(i) - XMEDIAN|]/N where the summation is from 1 to N

where XMEDIAN is the sample median.

Software

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