# InverseGaussianDistribution

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 • 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 (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

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## Basic Examples(6)

Probability density function:

 In:= Out= In:= Out= Cumulative distribution function of an inverse Gaussian distribution:

 In:= Out= In:= Out= Probability density function of a generalized inverse Gaussian distribution:

 In:= Out= In:= Out= Cumulative distribution function of a generalized inverse Gaussian distribution:

 In:= Out= Mean:

 In:= Out= Mean of a generalized inverse Gaussian distribution:

 In:= Out= Variance:

 In:= Out= Variance of a generalized inverse Gaussian distribution:

 In:= Out= ## Neat Examples(1)

Introduced in 2007
(6.0)
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Updated in 2016
(10.4)