dist.Laplace {LaplacesDemon} | R Documentation |

## Laplace Distribution: Univariate Symmetric

### Description

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

and scale
parameter `\lambda`

.

### Usage

```
dlaplace(x, location=0, scale=1, log=FALSE)
plaplace(q, location=0, scale=1)
qlaplace(p, location=0, scale=1)
rlaplace(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 |

`scale` |
This is the scale parameter |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \frac{1}{2 \lambda} \exp(-\frac{|\theta - \mu|}{\lambda})`

Inventor: Pierre-Simon Laplace (1774)

Notation 1:

`\theta \sim \mathrm{Laplace}(\mu,\lambda)`

Notation 2:

`\theta \sim \mathcal{L}(\mu, \lambda)`

Notation 3:

`p(\theta) = \mathrm{Laplace}(\theta | \mu, \lambda)`

Notation 4:

`p(\theta) = \mathcal{L}(\theta | \mu, \lambda)`

Parameter 1: location parameter

`\mu`

Parameter 2: scale parameter

`\lambda > 0`

Mean:

`E(\theta) = \mu`

Variance:

`var(\theta) = 2 \lambda^2`

Mode:

`mode(\theta) = \mu`

The Laplace distribution (Laplace, 1774) 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.

There are many extensions to the Laplace distribution, such as the asymmetric Laplace, asymmetric log-Laplace, Laplace (re-parameterized for precision), log-Laplace, multivariate Laplace, and skew-Laplace, among many more.

These functions are similar to those in the `VGAM`

package.

### Value

`dlaplace`

gives the density,
`plaplace`

gives the distribution function,
`qlaplace`

gives the quantile function, and
`rlaplace`

generates random deviates.

### References

Laplace, P. (1774). "Memoire sur la Probabilite des Causes par les
Evenements." l'Academie Royale des Sciences, 6, 621–656. English
translation by S.M. Stigler in 1986 as "Memoir on the Probability
of the Causes of Events" in *Statistical Science*, 1(3),
p. 359–378.

### See Also

`dalaplace`

,
`dallaplace`

,
`dexp`

,
`dlaplacep`

,
`dllaplace`

,
`dmvl`

,
`dnorm`

,
`dnormp`

,
`dnormv`

,
`dsdlaplace`

, and
`dslaplace`

.

### Examples

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

*LaplacesDemon*version 16.1.6 Index]