dist.Inverse.Gaussian | R Documentation |
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, |
lambda |
This is the inverse-variance parameter,
|
log |
Logical. If |
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.
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.
dnorm
,
dnormp
, and
dnormv
.
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"))
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