# InverseGaussianDistribution

InverseGaussianDistribution[μ,λ]

represents an inverse Gaussian distribution with mean μ and scale parameter λ.

InverseGaussianDistribution[μ,λ,θ]

represents a generalized inverse Gaussian distribution with parameters μ, λ, and θ.

# Details

- InverseGaussianDistribution[μ,λ] is also known as the inverse normal or Wald distribution.
- InverseGaussianDistribution[μ,λ,θ] is also known as the Sichel distribution.
- The probability density for value in an inverse Gaussian distribution is proportional to for , and zero for . »
- The probability density for value in a generalized inverse Gaussian distribution is proportional to for , and zero for .
- InverseGaussianDistribution allows μ and λ to be any positive real numbers and θ to be any real number.
- InverseGaussianDistribution allows λ and μ to be any quantities of the same unit dimensions, and θ to be a dimensionless quantity. »
- InverseGaussianDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

# Background & Context

- InverseGaussianDistribution[μ,λ,θ] represents a continuous statistical distribution defined over the interval and parametrized by a real number θ (called an "index parameter") and by two positive real numbers μ (the mean of the distribution) and λ (called a "scale parameter"). Overall, the probability density function (PDF) of an inverse Gaussian distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and its concentration near the axis) is determined by the values of μ, λ, and θ. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The three-parameter version is sometimes referred to as the generalized inverse Gaussian distribution or the Sichel distribution, while the two-parameter form InverseGaussianDistribution[μ,λ] (which is equivalent to InverseGaussianDistribution[μ,λ,-1/2]) is most often referred to as "the" inverse Gaussian distribution, though it is also sometimes referred to as Wald's distribution.
- Unlike the "inverse" relationships held between several other pairs of probability distributions (e.g. InverseGammaDistribution being inverse to GammaDistribution and InverseChiSquareDistribution being inverse to ChiSquareDistribution), and characterized by the behavior of certain reciprocals of random variables, InverseGaussianDistribution is not the distribution followed by the reciprocal 1/X of a normally distributed (or Gaussian) variate XNormalDistribution[μ,σ]. Instead, the term "inverse" in InverseGaussianDistribution refers to the fact that the time a Brownian motion with positive drift takes to reach a fixed positive level is distributed according to an inverse Gaussian distribution, while the Gaussian distribution describes the level of a Brownian motion at a fixed time. In addition to its role in Brownian motion, InverseGaussianDistribution is used as a tool in the study of Wiener processes, engineering, reliability theory, occupational exposure data, risk analysis, and actuarial statistics.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from an inverse Gaussian distribution. Distributed[x,InverseGaussianDistribution[μ,λ,θ]], written more concisely as xInverseGaussianDistribution[μ,λ,θ], can be used to assert that a random variable x is distributed according to an inverse Gaussian distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for inverse Gaussian distributions may be given using PDF[InverseGaussianDistribution[μ,λ,θ],x] and CDF[InverseGaussianDistribution[μ,λ,θ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with an inverse Gaussian distribution, EstimatedDistribution to estimate an inverse Gaussian parametric distribution from given data, and FindDistributionParameters to fit data to an inverse Gaussian distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic inverse Gaussian distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic inverse Gaussian distribution.
- TransformedDistribution can be used to represent a transformed inverse Gaussian distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain an inverse Gaussian distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving inverse Gaussian distributions.
- InverseGaussianDistribution is closely related to a number of other distributions. For example, it is related to NormalDistribution in the sense that the CumulantGeneratingFunction for NormalDistribution is inverse to that of an unscaled InverseGaussianDistribution. InverseGaussianDistribution has GammaDistribution, InverseGammaDistribution, and HyperbolicDistribution as limiting cases. InverseGaussianDistribution is also related to WeibullDistribution, LogNormalDistribution, ChiSquareDistribution, InverseChiSquareDistribution, and FRatioDistribution.

# Examples

open allclose all## Basic Examples (6)

## Scope (10)

Generate a sample of random numbers from an inverse Gaussian distribution:

Compare its histogram to the PDF:

Generate a set of random numbers distributed according to a generalized inverse Gaussian distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

For generalized inverse Gaussian distribution:

For generalized inverse Gaussian distribution:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Moments for generalized inverse Gaussian distribution:

Closed form for symbolic order:

Hazard function of inverse Gaussian distribution:

Hazard function of generalized inverse Gaussian distribution:

Quantile function of inverse Gaussian distribution:

Quantile function of generalized inverse Gaussian distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

## Applications (3)

Find the distribution of the time a Brownian motion with positive drift takes to reach a level of 2:

Remove empty lists and extract times:

Fit InverseGaussianDistribution to the data:

Compare data histogram to the fitted PDF:

The lifetime of a device has an inverse Gaussian distribution. Find the reliability of the device:

Find the maximum failure rate:

The failure rate decreases to a positive value:

Find reliability of two such devices in series:

Find reliability of two such devices in parallel:

Compare reliability of both systems for and :

When a particle travels through a medium it loses energy through scattering. The energy-loss spectrum, according to an integrable model by Lindhard and Nielsen, has an InverseGaussianDistribution profile:

The distribution is parametrized by its mean, which is proportional to medium thickness:

## Properties & Relations (6)

Scaling of inverse Gaussian distribution carries over the mean and the scale parameter:

Sum of independent identically distributed variables following InverseGaussianDistribution follows InverseGaussianDistribution:

Find the distribution of the sum using TransformedDistribution:

The mean of identical inverse Gaussian distributions has an inverse Gaussian distribution:

The sum of inverse Gaussian distribution variates with follows an inverse Gaussian distribution:

Relationships to other distributions:

Generalized inverse Gaussian distribution simplifies to inverse Gaussian distribution for :

## Possible Issues (2)

InverseGaussianDistribution is not defined when μ is not a positive real number:

InverseGaussianDistribution is not defined when λ is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: