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Exponential distribution


Exponential Distribution

Probability Density FunctionThe general formula for the probability density function of the exponential distribution isf(x) = (1/beta)*EXP(-(x - mu)/beta) for x >= mu; beta > 0
where mu is the location parameter and beta is the scale parameter(the scale parameter is often referred to as lambda which equals 1/beta). The case where mu = 0 and beta = 1 is called the standard exponential distribution. The equation for the standard exponential distribution is
f(x) = EXP(-x) for x >= 0
The general form of probability functions can be expressed in terms of the standard distribution. Subsequent formulas in this section are given for the 1-parameter (i.e., with scale parameter) form of the function.
The following is the plot of the exponential probability density function.
plot of the exponential probability density function
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the exponential distribution isF(x) = 1 - EXP(-x/beta) for x >= 0; beta > 0
The following is the plot of the exponential cumulative distribution function.
plot of the exponential cumulative distribution
Percent Point FunctionThe formula for the percent point function of the exponential distribution isG(p) = -beta*ln(1-p)
The following is the plot of the exponential percent point function.
plot of the exponential percent point function
Hazard FunctionThe formula for the hazard function of the exponential distribution ish(x) = lambda for x >= 0
The following is the plot of the exponential hazard function.
plot of the exponential hazard function
Cumulative Hazard FunctionThe formula for the cumulative hazard function of the exponential distribution isH(x) = x/beta for x >= 0; beta > 0
The following is the plot of the exponential cumulative hazard function.
plot of the exponential cumulative hazard function
Survival FunctionThe formula for the survival function of the exponential distribution isS(x) = EXP(-x/beta) for x>= 0; beta > 0
The following is the plot of the exponential survival function.
plot of the exponential survival function
Inverse Survival FunctionThe formula for the inverse survival function of the exponential distribution isG(p) = -beta*ln(p) 0 <= p < 1; beta > 0
The following is the plot of the exponential inverse survival function.
plot of the exponential inverse survival function
Common Statistics
Meanbeta
Medianbeta*ln(2)
ModeZero
RangeZero to plus infinity
Standard Deviationbeta
Coefficient of Variation1
Skewness2
Kurtosis9
Parameter EstimationFor the full sample case, the maximum likelihood estimator of the scale parameter is the sample mean. Maximum likelihood estimation for the exponential distribution is discussed in the chapter on reliability (Chapter 8). It is also discussed in chapter 19 of Johnson, Kotz, and Balakrishnan.
CommentsThe exponential distribution is primarily used in reliabilityapplications. The exponential distribution is used to model data with a constant failure rate (indicated by the hazard plot which is simply equal to a constant).
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the exponential distribution

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