# Exponential

##### The Exponential Distribution

Density, distribution function, quantile function and random
generation for the exponential distribution with rate `rate`

(i.e., mean `1/rate`

).

- Keywords
- distribution

##### Usage

```
dexp(x, rate = 1, log = FALSE)
pexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE)
qexp(p, rate = 1, lower.tail = TRUE, log.p = FALSE)
rexp(n, rate = 1)
```

##### Arguments

- x, q
- vector of quantiles.
- p
- vector of probabilities.
- n
- number of observations. If
`length(n) > 1`

, the length is taken to be the number required. - rate
- vector of rates.
- log, log.p
- logical; if TRUE, probabilities p are given as log(p).
- lower.tail
- logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

##### Details

If `rate`

is not specified, it assumes the default value of
`1`

.

The exponential distribution with rate $\lambda$ has density $$f(x) = \lambda {e}^{- \lambda x}$$ for $x \ge 0$.

##### Value

`dexp`

gives the density,`pexp`

gives the distribution function,`qexp`

gives the quantile function, and`rexp`

generates random deviates.The length of the result is determined by

`n`

for`rexp`

, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than`n`

are recycled to the length of the result. Only the first elements of the logical arguments are used.

##### Note

The cumulative hazard $H(t) = - \log(1 - F(t))$
is `-pexp(t, r, lower = FALSE, log = TRUE)`

.

##### source

`dexp`

, `pexp`

and `qexp`

are all calculated
from numerically stable versions of the definitions.

`rexp`

uses

Ahrens, J. H. and Dieter, U. (1972).
Computer methods for sampling from the exponential and normal distributions.
*Communications of the ACM*, **15**, 873--882.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
*Continuous Univariate Distributions*, volume 1, chapter 19.
Wiley, New York.

##### See Also

`exp`

for the exponential function.

Distributions for other standard distributions, including
`dgamma`

for the gamma distribution and
`dweibull`

for the Weibull distribution, both of which
generalize the exponential.

##### Examples

`library(stats)`

```
dexp(1) - exp(-1) #-> 0
## a fast way to generate *sorted* U[0,1] random numbers:
rsunif <- function(n) { n1 <- n+1
cE <- cumsum(rexp(n1)); cE[seq_len(n)]/cE[n1] }
plot(rsunif(1000), ylim=0:1, pch=".")
abline(0,1/(1000+1), col=adjustcolor(1, 0.5))
```

*Documentation reproduced from package stats, version 3.3, License: Part of R 3.3*