R: Inverse Gaussian Distribution

dist.Inverse.Gaussian {LaplacesDemon} | R Documentation

## 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) = (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

    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"))

[Package LaplacesDemon version 16.1.1 Index]