LaplacesDemon (version 16.1.1)

dist.Log.Normal.Precision: Log-Normal Distribution: Precision Parameterization

Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate log-normal distribution with mean \(\mu\) and precision \(\tau\).

Usage

dlnormp(x, mu, tau=NULL, var=NULL, log=FALSE)
plnormp(q, mu, tau, lower.tail=TRUE, log.p=FALSE)
qlnormp(p, mu, tau, lower.tail=TRUE, log.p=FALSE)
rlnormp(n, mu, tau=NULL, var=NULL)

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 mean parameter \(\mu\).

tau

This is the precision parameter \(\tau\), which must be positive. Tau and var cannot be used together

var

This is the variance parameter, which must be positive. Tau and var cannot be used together

log, log.p

Logical. If TRUE, then probabilities \(p\) are given as \(\log(p)\).

lower.tail

Logical. If TRUE (default), then probabilities are \(Pr[X \le x]\), otherwise, \(Pr[X > x]\).

Value

dlnormp gives the density, plnormp gives the distribution function, qlnormp gives the quantile function, and rlnormp generates random deviates.

Details

• Application: Continuous Univariate

• Density: \(p(\theta) = \sqrt{\frac{\tau}{2\pi}} \frac{1}{\theta} \exp(-\frac{\tau}{2} (\log(\theta - \mu))^2)\)

• Inventor: Carl Friedrich Gauss or Abraham De Moivre

• Notation 1: \(\theta \sim \mathrm{Log-}\mathcal{N}(\mu, \tau^{-1})\)

• Notation 2: \(p(\theta) = \mathrm{Log-}\mathcal{N}(\theta | \mu, \tau^{-1})\)

• Parameter 1: mean parameter \(\mu\)

• Parameter 2: precision parameter \(\tau > 0\)

• Mean: \(E(\theta) = \exp(\mu + \tau^{-1} / 2)\)

• Variance: \(var(\theta) = (\exp(\tau^{-1}) - 1)\exp(2\mu + \tau^{-1})\)

• Mode: \(mode(\theta) = \exp(\mu - \tau^{-1})\)

The log-normal distribution, also called the Galton distribution, is applied to a variable whose logarithm is normally-distributed. The distribution is usually parameterized with mean and variance, or in Bayesian inference, with mean and precision, where precision is the inverse of the variance. In contrast, Base R parameterizes the log-normal distribution with the mean and standard deviation. These functions provide the precision parameterization for convenience and familiarity.

A flat distribution is obtained in the limit as \(\tau \rightarrow 0\).

These functions are similar to those in base R.

See Also

dnorm, dnormp, dnormv, and prec2var.

Examples

# NOT RUN {
library(LaplacesDemon)
x <- dlnormp(1,0,1)
x <- plnormp(1,0,1)
x <- qlnormp(0.5,0,1)
x <- rlnormp(100,0,1)

#Plot Probability Functions
x <- seq(from=0.1, to=3, by=0.01)
plot(x, dlnormp(x,0,0.1), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dlnormp(x,0,1), type="l", col="green")
lines(x, dlnormp(x,0,5), type="l", col="blue")
legend(2, 0.9, expression(paste(mu==0, ", ", tau==0.1),
     paste(mu==0, ", ", tau==1), paste(mu==0, ", ", tau==5)),
     lty=c(1,1,1), col=c("red","green","blue"))
# }