# dist.Inverse.Gaussian: Inverse Gaussian Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

This is the density function and random generation from the inverse gaussian distribution.

## Usage

 1 2 dinvgaussian(x, mu, lambda, log=FALSE) rinvgaussian(n, mu, lambda) 

## Arguments

 n This is the number of draws from the distribution. x This is the scalar location to evaluate density. mu This is the mean parameter, mu. lambda This is the inverse-variance parameter, lambda. log Logical. If log=TRUE, then the logarithm of the density is returned.

## Details

• Application: Continuous Univariate

• Density: p(theta) = (lambda / (2*pi*theta^3))^(1/2) * exp(-((lambda*(theta-mu)^2) / (2*mu^2*theta))), theta > 0

• Inventor: Schrodinger (1915)

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

• Notation 2: p(theta) = N^-1(theta | mu, lambda)

• Parameter 1: shape mu > 0

• Parameter 2: scale lambda > 0

• Mean: E(theta) = mu

• Variance: var(theta) = mu^3/lambda

• Mode: mode(theta) = mu*((1 + ((9*mu^2)/(4*lambda^2)))^(1/2) - \frac{3*mu}{2*lambda})

The inverse-Gaussian distribution, also called the Wald distribution, is used when modeling dependent variables that are positive and continuous. When lambda tends to infinity (or variance to zero), the inverse-Gaussian distribution becomes similar to a normal (Gaussian) distribution. The name, inverse-Gaussian, is misleading, because it is not the inverse of a Gaussian distribution, which is obvious from the fact that theta must be positive.

## Value

dinvgaussian gives the density and rinvgaussian generates random deviates.

## References

Schrodinger E. (1915). "Zur Theorie der Fall-und Steigversuche an Teilchenn mit Brownscher Bewegung". Physikalische Zeitschrift, 16, p. 289–295.

## See Also

dnorm, dnormp, and dnormv.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 library(LaplacesDemon) x <- dinvgaussian(2, 1, 1) x <- rinvgaussian(10, 1, 1) #Plot Probability Functions x <- seq(from=1, to=20, by=0.1) plot(x, dinvgaussian(x,1,0.5), ylim=c(0,1), type="l", main="Probability Function", ylab="density", col="red") lines(x, dinvgaussian(x,1,1), type="l", col="green") lines(x, dinvgaussian(x,1,5), type="l", col="blue") legend(2, 0.9, expression(paste(mu==1, ", ", sigma==0.5), paste(mu==1, ", ", sigma==1), paste(mu==1, ", ", sigma==5)), lty=c(1,1,1), col=c("red","green","blue")) 

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.