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

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dlnormp(x, mu, tau, 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)

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.

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 <= x], otherwise, Pr[X > x].

Details

  • Application: Continuous Univariate

  • Density: p(theta) = sqrt(tau/(2*pi)) * (1/theta) * exp(-(tau/2)*(log(theta-mu))^2)

  • Inventor: Carl Friedrich Gauss or Abraham De Moivre

  • Notation 1: theta ~ Log-N(mu, tau^(-1))

  • Notation 2: p(theta) = Log-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 -> 0.

These functions are similar to those in base R.

Value

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

Author(s)

Statisticat, LLC. software@bayesian-inference.com

See Also

dnorm, dnormp, dnormv, and prec2var.

Examples

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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"))

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