# 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)`

.

##### 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)`

```
# NOT RUN {
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 3.6.2, License: Part of R 3.6.2*