dist.Inverse.Gaussian {LaplacesDemon}

Inverse Gaussian Distribution

Description

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

Usage

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) = \frac{\lambda}{(2 \pi \theta^3)^{1/2}} \exp(-\frac{\lambda (\theta - \mu)^2}{2 \mu^2 \theta}), \theta > 0

• Inventor: Schrodinger (1915)

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

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

• Parameter 1: shape \mu > 0

• Parameter 2: scale \lambda > 0

• Mean: E(\theta) = \mu

• Variance: var(\theta) = \frac{\mu^3}{\lambda}

• Mode: mode(\theta) = \mu((1 + \frac{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 \rightarrow \infty (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

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"))