dist.Skew.Discrete.Laplace {LaplacesDemon} | R Documentation |

## Skew Discrete Laplace Distribution: Univariate

### 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

```
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 |

`q` |
This is a scalar or vector of parameter |

`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 |

### Details

Application: Discrete Univariate

Density 1:

`p(\theta) = \frac{(1-p)(1-q)}{1-pq}p^\theta; \theta=0,1,2,3,\dots`

Density 2:

`p(\theta) = \frac{(1-p)(1-q)}{1-pq}q^{|\theta|}; x=0,-1,-2,-3,\dots`

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

Notation 1:

`\theta \sim \mathcal{DL}(p, q)`

Notation 2:

`p(\theta) = \mathcal{DL}(\theta | p, q)`

Parameter 1:

`p \in [0,1]`

Parameter 2:

`q \in [0,1]`

Mean 1:

`E(\theta) = \frac{1}{1-p}-\frac{1}{1-q}=\frac{p}{1-p}-\frac{q}{1-q}`

Mean 2:

`E(|\theta|) = \frac{q(1-p)^2+p(1-q)^2}{(1-qp)(1-q)(1-p)}`

Variance:

`var(\theta) = \frac{1}{(1-p)^2(1-q)^2}[\frac{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.

### See Also

`dalaplace`

,
`dexp`

,
`dlaplace`

,
`dlaplacep`

, and
`dslaplace`

.

### Examples

```
library(LaplacesDemon)
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"))
```

*LaplacesDemon*version 16.1.6 Index]