VGAM (version 0.7-1)

laplaceUC: 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)
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 $\mu$, which is the mean.

scale

the scale parameter $b$. Must consist of positive values.

Value

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

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 $$f(y) = \frac{1}{2b} \exp \left( - \frac{|y-\mu|}{b} \right)$$ where $-\infty0$. The mean is $\mu$ and the variance is $2b^2$.

References

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

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


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)

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