# dist.Log.Laplace: Log-Laplace Distribution: Univariate Symmetric

### Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, symmetric, log-Laplace distribution with location parameter `location` and scale parameter `scale`.

### Usage

 ```1 2 3 4``` ```dllaplace(x, location=0, scale=1, log=FALSE) pllaplace(q, location=0, scale=1) qllaplace(p, location=0, scale=1) rllaplace(n, location=0, scale=1) ```

### Arguments

 `x, q` These are each a vector of quantiles. `p` This is a vector of probabilities. `n` This is the number of observations, which must be a positive integer that has length 1. `location` This is the location parameter mu. `scale` This is the scale parameter lambda, which must be positive. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

• Application: Continuous Univariate

• Density 1: p(theta) = ((sqrt(2)/lambda)^2 / (2*(sqrt(2)/lambda))) * exp(-(sqrt(2)/lambda)*(theta - mu)), theta >= exp(mu)

• Density 2: p(theta) = ((sqrt(2)/lambda)^2 / (2*(sqrt(2)/lambda))) * exp((sqrt(2)/lambda)*(theta - mu)),theta < exp(mu)

• Inventor: Pierre-Simon Laplace

• Notation 1: theta ~ LL(mu, lambda)

• Notation 2: p(theta) = LL(theta | mu, lambda)

• Parameter 1: location parameter mu

• Parameter 2: scale parameter lambda > 0

• Mean: E(theta) =

• Variance: var(theta) =

• Mode: mode(theta) =

The univariate, symmetric log-Laplace distribution is derived from the Laplace distribution. Multivariate and asymmetric versions also exist.

These functions are similar to those in the `VGAM` package.

### Value

`dllaplace` gives the density, `pllaplace` gives the distribution function, `qllaplace` gives the quantile function, and `rllaplace` generates random deviates.

### References

Kozubowski, T. J. and Podgorski, K. (2003). "Log-Laplace Distributions". International Mathematical Journal, 3, p. 467–495.

`dalaplace`, `dallaplace`, `dexp`, `dlaplace`, `dlaplacep`, `dmvl`, `dnorm`, `dnormp`, and `dnormv`.

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```library(LaplacesDemon) x <- dllaplace(1,0,1) x <- pllaplace(1,0,1) x <- qllaplace(0.5,0,1) x <- rllaplace(100,0,1) #Plot Probability Functions x <- seq(from=0.1, to=20, by=0.1) plot(x, dllaplace(x,0,0.1), ylim=c(0,1), type="l", main="Probability Function", ylab="density", col="red") lines(x, dllaplace(x,0,0.5), type="l", col="green") lines(x, dllaplace(x,0,1.5), type="l", col="blue") legend(2, 0.9, expression(paste(mu==0, ", ", lambda==0.1), paste(mu==0, ", ", lambda==0.5), paste(mu==0, ", ", lambda==1.5)), lty=c(1,1,1), col=c("red","green","blue")) ```

Search within the LaplacesDemon package
Search all R packages, documentation and source code

Questions? Problems? Suggestions? or email at ian@mutexlabs.com.

Please suggest features or report bugs with the GitHub issue tracker.

All documentation is copyright its authors; we didn't write any of that.