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Double exponential distribution

Double Exponential Distribution

Probability Density FunctionThe general formula for the probability density function of the double exponential distribution isf(x) = EXP(-|(x-mu)/beta|)/(2*beta)
where mu is the location parameter and beta is the scale parameter. The case where mu = 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 FunctionThe formula for the cumulative distribution function of the double exponential distribution isF(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 FunctionThe formula for the percent point function of the double exponential distribution isG(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 FunctionThe formula for the hazard function of the double exponential distribution ish(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 FunctionThe formula for the cumulative hazard function of the double exponential distribution isH(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 FunctionThe 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 FunctionThe formula for the inverse survival function of the double exponential distribution isZ(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
Meanmu
Medianmu
Modemu
RangeNegative infinity to positive infinity
Standard DeviationSQRT(2)*beta
Skewness0
Kurtosis6
Coefficient of VariationSQRT(2)*(beta/mu)
Parameter EstimationThe maximum likelihood estimators of the location and scale parameters of the double exponential distribution aremuhat = XMEDIAN
betahat = SUM[|X(i) - XMEDIAN|]/N where the summation is from 1 to N
where XMEDIAN is the sample median.
SoftwareSome general purpose statistical software programs support at least some of the probability functions for the double exponential distribution.

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