R: Log-Normal Distribution: Precision Parameterization

dist.Log.Normal.Precision {LaplacesDemon} | R Documentation

## 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 <= 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

    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"))

[Package LaplacesDemon version 16.1.1 Index]