BUILT-IN MATHEMATICA SYMBOL

InverseGaussianDistribution

represents an inverse Gaussian distribution with mean

and scale parameter

.

InverseGaussianDistribution

represents a generalized inverse Gaussian distribution with parameters

,

, and

.

MORE INFORMATION

InverseGaussianDistribution is also known as the inverse normal or Wald distribution.

InverseGaussianDistribution 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 and to be any positive real numbers and to be any real number.

InverseGaussianDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

EXAMPLES

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Basic Examples (6)

Probability density function:

Cumulative distribution function of an inverse Gaussian distribution:

Probability density function of a generalized inverse Gaussian distribution:

Cumulative distribution function of a generalized inverse Gaussian distribution:

Mean:

Mean of a generalized inverse Gaussian distribution:

Variance:

Variance of a generalized inverse Gaussian distribution:

Probability density function:

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Cumulative distribution function of an inverse Gaussian distribution:

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Probability density function of a generalized inverse Gaussian distribution:

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Cumulative distribution function of a generalized inverse Gaussian distribution:

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Mean:

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Mean of a generalized inverse Gaussian distribution:

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Variance:

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Variance of a generalized inverse Gaussian distribution:

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Scope (9)

Generate a set of random numbers distributed according to 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:

Skewness:

For generalized inverse Gaussian distribution:

Kurtosis:

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:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

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:

Applications (2)

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

The failure rate for

and

:

The failure rate has a maximum:

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 parameterized by its mean, which is proportional to medium thickness:

Probability density:

Properties & Relations (7)

Parameter influence on the CDF for each

:

Generalized inverse Gaussian distribution:

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

Sum of

independent identically distributed variables following InverseGaussianDistribution follows InverseGaussianDistribution:

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

is not a positive real number:

InverseGaussianDistribution is not defined when

is not a positive real number:

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