# The Laplace Distribution Density, distribution function, quantile function and random generation for the Laplace distribution with location parameter `location` and scale parameter `scale`. ```r dlaplace(x, location = 0, scale = 1, log = FALSE) plaplace(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) qlaplace(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) rlaplace(n, location = 0, scale = 1) ``` ## Arguments - `x, q`: vector of quantiles. - `p`: vector of probabilities. - `n`: number of observations. Same as in `runif`. - `location`: the location parameter $a$, which is the mean. - `scale`: the scale parameter $b$. Must consist of positive values. - `log`: Logical. If `log = TRUE` then the logarithm of the density is returned. - `lower.tail, log.p`: Same meaning as in `pnorm` or `qnorm`. ## Details The Laplace distribution is often known as the double-exponential distribution and, for modelling, has heavier tail than the normal distribution. The Laplace density function is $$ f(y) = \frac{1}{2b} \exp \left( - \frac{|y-a|}{b}\right)%f(y) = (1/(2b)) exp( -|y-a|/b ) $$ where $-Inf0$. The mean is $a$ and the variance is $2b^2$. See `laplace`, the `VGAM` family function for estimating the two parameters by maximum likelihood estimation, for formulae and details. Apart from `n`, all the above arguments may be vectors and are recyled to the appropriate length if necessary. ## Returns `dlaplace` gives the density, `plaplace` gives the distribution function, `qlaplace` gives the quantile function, and `rlaplace` generates random deviates. ## References Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). **Statistical Distributions**, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition. ## Author(s) T. W. Yee and Kai Huang ## See Also `laplace`. ## Examples ```r loc <- 1; b <- 2 y <- rlaplace(n = 100, loc = loc, scale = b) mean(y) # sample mean loc # population mean var(y) # sample variance 2 * b^2 # population variance ## Not run: loc <- 0; b <- 1.5; x <- seq(-5, 5, by = 0.01) plot(x, dlaplace(x, loc, b), type = "l", col = "blue", main = "Blue is density, orange is the CDF", ylim = c(0,1), sub = "Purple are 5,10,...,95 percentiles", las = 1, ylab = "") abline(h = 0, col = "blue", lty = 2) lines(qlaplace(seq(0.05,0.95,by = 0.05), loc, b), dlaplace(qlaplace(seq(0.05, 0.95, by = 0.05), loc, b), loc, b), col = "purple", lty = 3, type = "h") lines(x, plaplace(x, loc, b), type = "l", col = "orange") abline(h = 0, lty = 2) ## End(Not run) plaplace(qlaplace(seq(0.05, 0.95, by = 0.05), loc, b), loc, b) ```

laplaceUC function