The double-exponential distribution

Arguments

x, q

vector of quantiles.

rate

vector of rates.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Value

ddexp() gives the density, pdexp() gives the distribution function, qdexp() gives the quantile function, and rdexp() generates random deviates.

Details

If rate is not specified, it assumes the default value of 1.

The double-exponential distribution with rate $\lambda$ has density $$f(x) =\frac{\lambda}{2}e^{-\lambda | x |}$$ for $x \in \mathbb{R}$. The cummulative distribution is: $$F(x) = \int_{-\infty}^{x}f(t) dt =\left\{ \begin{array}{ll} \frac{1}{2}e^{\lambda x} & \text{if } x < 0\\ 1-\frac{1}{2}e^{-\lambda x} & \text{if } x \geq 0 \end{array} \ \right.$$ The inverse cumulative distribution function is given by $$F^{-1}(p) = - \operatorname{sign}(p-0.5)\frac{\ln(1 - 2 | p - 0.5 | )} {\lambda}.$$

See also

Examples

set.seed(54321)
rate <- 2
rx <- rdexp(10^3, rate)
hist(rx, breaks = "FD", freq = FALSE)
lines(density(rx), lty = 2, col = "blue")
curve(ddexp(x, rate), lwd = 2, add = TRUE)
p <- c(0.005, 0.025, 0.05, 0.1)
abline(v = qdexp(c(p, 1-p), rate = rate), lty = 3, col = "red")

plot(ecdf(rx))
curve(pdexp(x, rate), lwd = 2, add = TRUE)