dist.Laplace.Precision {LaplacesDemon} | R Documentation |

## Laplace Distribution: Precision Parameterization

### Description

These functions provide the density, distribution function, quantile
function, and random generation for the univariate, symmetric, Laplace
distribution with location parameter `\mu`

and precision
parameter `\tau`

, which is the inverse of the usual scale
parameter, `\lambda`

.

### Usage

```
dlaplacep(x, mu=0, tau=1, log=FALSE)
plaplacep(q, mu=0, tau=1)
qlaplacep(p, mu=0, tau=1)
rlaplacep(n, mu=0, tau=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. |

`mu` |
This is the location parameter |

`tau` |
This is the precision parameter |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \frac{\tau}{2} \exp(-\tau |\theta-\mu|)`

Inventor: Pierre-Simon Laplace (1774)

Notation 1:

`\theta \sim \mathrm{Laplace}(\mu,\tau^{-1})`

Notation 2:

`\theta \sim \mathcal{L}(\mu, \tau^{-1})`

Notation 3:

`p(\theta) = \mathrm{Laplace}(\mu,\tau^{-1})`

Notation 4:

`p(\theta) = \mathcal{L}(\theta | \mu, \tau^{-1})`

Parameter 1: location parameter

`\mu`

Parameter 2: precision parameter

`\tau > 0`

Mean:

`E(\theta) = \mu`

Variance:

`var(\theta) = 2 \tau^{-2}`

Mode:

`mode(\theta) = \mu`

The Laplace distribution is also called the double exponential
distribution, because it looks like two exponential distributions back to
back with respect to location `\mu`

. It is also called the
“First Law of Laplace”, just as the normal distribution is referred to
as the “Second Law of Laplace”. The Laplace distribution is
symmetric with respect to `\mu`

, though there are asymmetric
versions of the Laplace distribution. The PDF of the Laplace
distribution is reminiscent of the normal distribution; however,
whereas the normal distribution is expressed in terms of the squared
difference from the mean `\mu`

, the Laplace density is
expressed in terms of the absolute difference from the mean,
`\mu`

. Consequently, the Laplace distribution has fatter
tails than the normal distribution. It has been argued that the Laplace
distribution fits most things in nature better than the normal
distribution. Elsewhere, there are a large number of extensions to the
Laplace distribution, including asymmetric versions and
multivariate versions, among many more. These functions provide the
precision parameterization for convenience and familiarity in Bayesian
inference.

### Value

`dlaplacep`

gives the density,
`plaplacep`

gives the distribution function,
`qlaplacep`

gives the quantile function, and
`rlaplacep`

generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`dalaplace`

,
`dexp`

,
`dlaplace`

,
`dmvl`

,
`dnorm`

,
`dnormp`

, and
`dnormv`

.

### Examples

```
library(LaplacesDemon)
x <- dlaplacep(1,0,1)
x <- plaplacep(1,0,1)
x <- qlaplacep(0.5,0,1)
x <- rlaplacep(100,0,1)
#Plot Probability Functions
x <- seq(from=-5, to=5, by=0.1)
plot(x, dlaplacep(x,0,0.5), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dlaplacep(x,0,1), type="l", col="green")
lines(x, dlaplacep(x,0,2), type="l", col="blue")
legend(2, 0.9, expression(paste(mu==0, ", ", tau==0.5),
paste(mu==0, ", ", tau==1), paste(mu==0, ", ", tau==2)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]