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

subject to the following conditions:

The exponential random variable 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 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.

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

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

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

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

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

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

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

$b$

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

$b^2$

$Y≔\mathrm{ExponentialRandomVariable}\left(3\right)\&colon;$

$\mathrm{PDF}\left(Y\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{CDF}\left(Y\&comma;x\right)$

$\left\{\begin{array}{cc}0& x<0\\ 1-{\&ExponentialE;}^{-\frac{x}{3}}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{CDF}\left(Y\&comma;4\&comma;\mathrm{output}=\mathrm{plot}\right)$

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.

The Student[Statistics][ExponentialRandomVariable] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.