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Statistics[Distributions]

 Exponential
 exponential distribution

 Calling Sequence Exponential(b) ExponentialDistribution(b)

Parameters

 b - scale parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{{ⅇ}^{-\frac{t}{b}}}{b}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Exponential}\left(b\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{{ⅇ}}^{{-}\frac{{u}}{{b}}}}{{b}}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{{ⅇ}}^{{-}\frac{{0.5}}{{b}}}}{{b}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${b}$ (3)

References

 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.