distributional (version 0.6.0)

dist_inverse_gaussian: The Inverse Gaussian distribution

Description

Usage

dist_inverse_gaussian(mean, shape)

Arguments

mean, shape

parameters. Must be strictly positive. Infinite values are supported.

Details

The inverse Gaussian distribution (also known as the Wald distribution) is commonly used to model positive-valued data, particularly in contexts involving first passage times and reliability analysis.

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_inverse_gaussian.html

In the following, let \(X\) be an Inverse Gaussian random variable with parameters mean = \(\mu\) and shape = \(\lambda\).

Support: \((0, \infty)\)

Mean: \(\mu\)

Variance: \(\frac{\mu^3}{\lambda}\)

Probability density function (p.d.f):

$$ f(x) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\left(-\frac{\lambda(x - \mu)^2}{2\mu^2 x}\right) $$

Cumulative distribution function (c.d.f):

$$ F(x) = \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu} - 1\right)\right) + \exp\left(\frac{2\lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu} + 1\right)\right) $$

where \(\Phi\) is the standard normal c.d.f.

Moment generating function (m.g.f):

$$ E(e^{tX}) = \exp\left(\frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2\mu^2 t}{\lambda}}\right)\right) $$

for \(t < \frac{\lambda}{2\mu^2}\).

Skewness: \(3\sqrt{\frac{\mu}{\lambda}}\)

Excess Kurtosis: \(\frac{15\mu}{\lambda}\)

Quantiles: No closed-form expression, approximated numerically.

See Also

actuar::InverseGaussian

Examples

dist <- dist_inverse_gaussian(mean = c(1,1,1,3,3), shape = c(0.2, 1, 3, 0.2, 1))
dist

if (FALSE) { # requireNamespace("actuar", quietly = TRUE)
mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
}