extraDistr (version 1.9.1)

DiscreteLaplace: Discrete Laplace distribution

Description

Probability mass, distribution function and random generation for the discrete Laplace distribution parametrized by location and scale.

Usage

ddlaplace(x, location, scale, log = FALSE)
pdlaplace(q, location, scale, lower.tail = TRUE, log.p = FALSE)
rdlaplace(n, location, scale)

Arguments

x, q

vector of quantiles.

location

location parameter.

scale

scale parameter; 0 < scale < 1.

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, \(P[X > x]\).

n

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

Details

If \(U \sim \mathrm{Geometric}(1-p)\) and \(V \sim \mathrm{Geometric}(1-p)\), then \(U-V \sim \mathrm{DiscreteLaplace}(p)\), where geometric distribution is related to discrete Laplace distribution in similar way as exponential distribution is related to Laplace distribution.

Probability mass function

$$ f(x) = \frac{1-p}{1+p} p^{|x-\mu|} $$

Cumulative distribution function

$$ F(x) = \left\{\begin{array}{ll} \frac{p^{-|x-\mu|}}{1+p} & x < 0 \\ 1 - \frac{p^{|x-\mu|+1}}{1+p} & x \ge 0 \end{array}\right. $$

References

Inusah, S., & Kozubowski, T.J. (2006). A discrete analogue of the Laplace distribution. Journal of statistical planning and inference, 136(3), 1090-1102.

Kotz, S., Kozubowski, T., & Podgorski, K. (2012). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer Science & Business Media.

Examples

p <- 0.45
x <- rdlaplace(1e5, 0, p)
xx <- seq(-200, 200, by = 1)
plot(prop.table(table(x)))
lines(xx, ddlaplace(xx, 0, p), col = "red")
hist(pdlaplace(x, 0, p))
plot(ecdf(x))
lines(xx, pdlaplace(xx, 0, p), col = "red", type = "s")