# dist.Skew.Discrete.Laplace: Skew Discrete Laplace Distribution: Univariate In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

## Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, skew discrete Laplace distribution with parameters p and q.

## Usage

 ```1 2 3 4``` ```dsdlaplace(x, p, q, log=FALSE) psdlaplace(x, p, q) qsdlaplace(prob, p, q) rsdlaplace(n, p, q) ```

## Arguments

 `x` This is a vector of data. `p` This is a scalar or vector of parameter p in [0,1]. `q` This is a scalar or vector of parameter q in [0,1]. `prob` This is a probability scalar or vector. `n` This is the number of observations, which must be a positive integer that has length 1. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned.

## Details

• Application: Discrete Univariate

• Density 1: p(theta) = (1-p)(1-q) / (1-pq)p^theta, theta=0,1,2,3,...

• Density 2: p(theta) = (1-p)(1-q) / (1-pq)q^(|theta|),; x=0,-1,-2,-3,...

• Inventor: Kozubowski, T.J. and Inusah, S. (2006)

• Notation 1: theta ~ DL(p, q)

• Notation 2: p(theta) = DL(theta | p, q)

• Parameter 1: p in [0,1]

• Parameter 2: q in [0,1]

• Mean 1: E(theta) = (1 / (1-p)) - (1 / (1-q)) = (p / (1-p)) - (q / (1-q))

• Mean 2: E(|theta|) = (q(1-p)^2+p(1-q)^2) / ((1-qp)(1-q)(1-p))

• Variance: var(theta) = (1 / ((1-p)^2(1-q)^2))[(q(1-p)^3(1+q)+p(1-q)^3(1+p)) / (1-pq) - (p-q)^2]

• Mode:

This is a discrete form of the skew-Laplace distribution. The symmetric discrete Laplace distribution occurs when p=q. DL(p,0) is a geometric distribution, and DL(0,q) is a geometric distribution of non-positive integers. The distribution is degenerate when DL(0,0). Since the geometric distribution is a discrete analog of the exponential distribution, the distribution of the difference of two geometric variables is a discrete Laplace distribution.

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

## Value

`dslaplace` gives the density, `pslaplace` gives the distribution function, `qslaplace` gives the quantile function, and `rslaplace` generates random deviates.

## References

Kozubowski, T.J. and Inusah, S. (2006). "A Skew Laplace Distribution on Integers". AISM, 58, p. 555–571.

`dalaplace`, `dexp`, `dlaplace`, `dlaplacep`, and `dslaplace`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```library(LaplacesDemonCpp) x <- dsdlaplace(1,0.5,0.5) x <- psdlaplace(1,0.5,0.5) x <- qsdlaplace(0.5,0.5,0.5) x <- rsdlaplace(5,0.5,0.5) #Plot Probability Functions x <- c(-3:3) plot(x, dsdlaplace(x,0.5,0.5), ylim=c(0,0.6), type="l", main="Probability Function", ylab="density", col="red") lines(x, dsdlaplace(x,0.3,0.6), type="l", col="green") lines(x, dsdlaplace(x,0.9,0.1), type="l", col="blue") legend(-2.5, 0.5, expression(paste(p==0.5, ", ", q==0.5), paste(p==0.3, ", ", q==0.6), paste(p==0.9, ", ", q==0.1)), lty=c(1,1,1), col=c("red","green","blue")) ```