dist.Log.Normal.Precision: Log-Normal Distribution: Precision Parameterization In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

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

 ```1 2 3 4``` ```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. [email protected]

`dnorm`, `dnormp`, `dnormv`, and `prec2var`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```library(LaplacesDemonCpp) 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")) ```