Discrete Laplace distribution
Probability mass, distribution function and random generation for the discrete Laplace distribution parametrized by location and scale.
ddlaplace(x, location, scale, log = FALSE) pdlaplace(q, location, scale, lower.tail = TRUE, log.p = FALSE) rdlaplace(n, location, scale)
x, q |
vector of quantiles. |
location |
location parameter. |
scale |
scale parameter; |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
n |
number of observations. If |
If U ~ Geometric(1-p) and V ~ Geometric(1-p), then U-V ~ 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) = (1-p)/(1+p) * p^(|x-μ|)
Cumulative distribution function
F(x) = [if x < 0:] p^-floor(x-μ))/(1+p) [else:] 1-(p^(floor(x-μ)+1))/(1+p)
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.
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")
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