Inverse Gaussian Distribution
This is the density function and random generation from the inverse gaussian distribution.
dinvgaussian(x, mu, lambda, log=FALSE) rinvgaussian(n, mu, lambda)
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 |
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.
dinvgaussian gives the density and
rinvgaussian generates random deviates.
Schrodinger E. (1915). "Zur Theorie der Fall-und Steigversuche an Teilchenn mit Brownscher Bewegung". Physikalische Zeitschrift, 16, p. 289–295.
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"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.