The exponential distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The exponential distribution has the lack of memory property: the probability of an event occurring in the next time interval of an exponential distribution is independent of the amount of time that has already passed.

The exponential variate with scale parameter b is a special case of the Gamma variate with scale parameter b and shape parameter 1: Exponential(b) ~ Gamma(b,1)

The exponential variate with scale parameter b is a special case of the Weibull variate with scale parameter b and shape parameter 1: Exponential(b) ~ Weibull(b,1)

The exponential variate with scale parameter b is related to the unit Uniform variate by the formula:  Exponential(b) ~ -b * log(Uniform(0,1))

The discrete analog of the exponential variate is the Geometric variate.

The exponential variate with scale parameter b is related to the Laplace variate with location parameter a and scale parameter b according to the formula:  Exponential(b) ~ abs(Laplace(a,b) - a).

Note that the Exponential command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Exponential}\left(b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{{\&ExponentialE;}^{-\frac{u}{b}}}{b}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{{\&ExponentialE;}^{-\frac{0.5}{b}}}{b}$

$\mathrm{Mean}\left(X\right)$

$b$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.