laplaceUC {VGAM} | R Documentation

## The Laplace Distribution

### Description

Density, distribution function, quantile function and random generation for the Laplace distribution with location parameter `location` and scale parameter `scale`.

### 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` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. Positive integer of length 1.
`location` | the location parameter a, which is the mean.
`scale` | the scale parameter b. Must consist of positive values.
`log` | Logical. If `log=TRUE` then the logarithm of the density is returned.

### Details

The Laplace distribution is often known as the double-exponential distribution and, for modelling, has heavier tail than the normal distribution. The Laplace density function is

f(y) = (1/(2b)) exp( -|y-a|/b )

where -Inf<y<Inf, -Inf<a<Inf and b>0. The mean is a and the variance is 2b^2.

See `laplace`, the VGAM family function for estimating the two parameters by maximum likelihood estimation, for formulae and details. Apart from `n`, all the above arguments may be vectors and are recyled to the appropriate length if necessary.

### Value

`dlaplace` gives the density, `plaplace` gives the distribution function, `qlaplace` gives the quantile function, and `rlaplace` generates random deviates.

### Author(s)

T. W. Yee

### References

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

### See Also

`laplace`.

### Examples

    loc = 1; b = 2
    y = rlaplace(n=100, loc=loc, scale=b)
    mean(y)
    loc      # population mean
    var(y)
    2 * b^2  # population variance

    ## Not run:
    x = seq(-5, 5, by=0.01)
    loc = 0; b = 1.5
    plot(x, dlaplace(x, loc, b), type="l", col="blue", ylim=c(0,1),
         main="Blue is density, red is cumulative distribution function",
         sub="Purple are 5,10,...,95 percentiles", las=1, ylab="")
    abline(h=0, col="blue", lty=2)
    lines(qlaplace(seq(0.05,0.95,by=0.05), loc, b),
          dlaplace(qlaplace(seq(0.05,0.95,by=0.05), loc, b), loc, b),
          col="purple", lty=3, type="h")
    lines(x, plaplace(x, loc, b), type="l", col="red")
    abline(h=0, lty=2)
    ## End(Not run)

    plaplace(qlaplace(seq(0.05,0.95,by=0.05), loc, b), loc, b)

[Package VGAM version 0.7-9 Index]