This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL

# InverseGaussianDistribution

 InverseGaussianDistributionrepresents an inverse Gaussian distribution with mean and scale parameter . InverseGaussianDistributionrepresents a generalized inverse Gaussian distribution with parameters , , and .
• 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 .
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:
 Out[1]=
 Out[2]=

Cumulative distribution function of an inverse Gaussian distribution:
 Out[1]=
 Out[2]=

Probability density function of a generalized inverse Gaussian distribution:
 Out[1]=
 Out[2]=

Cumulative distribution function of a generalized inverse Gaussian distribution:
 Out[1]=

Mean:
 Out[1]=
Mean of a generalized inverse Gaussian distribution:
 Out[2]=

Variance:
 Out[1]=
Variance of a generalized inverse Gaussian distribution:
 Out[2]=
 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:
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:
 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:
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 :
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:
New in 6 | Last modified in 8