# laplaceUC: The Laplace Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

Density, distribution function, quantile function and random generation for the Laplace distribution with location parameter `location` and scale parameter `scale`.

## Usage

 ```1 2 3 4``` ```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) = (1/(2b)) exp( -|y-a|/b )

where -Inf<y<Inf, -Inf<a<Inf and b>0. 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.

## Value

`dlaplace` gives the density, `plaplace` gives the distribution function, `qlaplace` gives the quantile function, and `rlaplace` generates random deviates.

## Author(s)

T. W. Yee and Kai Huang

## References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

`laplace`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```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", ylim = c(0,1), main = "Blue is density, orange is cumulative distribution function", 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) ```