BUILT-IN MATHEMATICA SYMBOL

InverseGammaDistribution

represents an inverse gamma distribution with shape parameter

and scale parameter

.

InverseGammaDistribution

represents a generalized inverse gamma distribution with shape parameters

and

, scale parameter

, and location parameter

.

MORE INFORMATION

The inverse gamma distribution InverseGammaDistribution is the distribution followed by the inverse of a GammaDistribution distributed random variable. »

InverseGammaDistribution is equivalent to TransformedDistribution[1/x, xGammaDistribution[, 1/]].

InverseGammaDistribution is equivalent to TransformedDistribution[1/x, xGammaDistribution[, 1/, , ]].

InverseGammaDistribution allows , , and to be any positive real numbers and to be any real number.

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

EXAMPLES

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Basic Examples (8)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

Probability density function for the generalized inverse gamma distribution:

Cumulative distribution function for the generalized inverse gamma distribution:

Mean and variance of the generalized inverse gamma distribution:

Median:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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Probability density function for the generalized inverse gamma distribution:

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Cumulative distribution function for the generalized inverse gamma distribution:

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Mean and variance of the generalized inverse gamma distribution:

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Median:

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Scope (9)

Generate a set of pseudorandom numbers that are inverse gamma distributed:

Compare its histogram to the PDF:

Generate a set of pseudorandom numbers that have generalized inverse gamma distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness depends only on shape parameter

:

As

gets larger, the distribution becomes more symmetric:

The generalized case depends on both

and

:

Kurtosis depends only on shape parameter

:

The kurtosis approaches the kurtosis of NormalDistribution

as

approaches

:

The generalized case depends on both

and

:

Different moments with closed forms as functions of parameters:

Different moments of generalized inverse gamma distribution:

Hazard function:

Hazard function of generalized inverse gamma distribution:

Quantile function:

Generalized inverse gamma distribution:

Applications (1)

The present value of one-dollar stochastic perpetuity when the rate obeys a Wiener process with shift

and volatility

follows InverseGaussianDistribution:

Find the expected present value:

Compute the no-volatility limit:

Find the probability that the present value is smaller than the no-volatility limit:

Compute the probability when

and

:

Properties & Relations (9)

Parameter influence on the CDF for each

:

Generalized inverse gamma distribution:

Inverse gamma distribution is closed under scaling by a positive factor:

Generalized inverse gamma distribution is closed under translation and scaling by a positive factor:

Relationships to other distributions:

InverseChiSquareDistribution is a special case of inverse gamma distribution:

Generalized InverseChiSquareDistribution is a special case of inverse gamma distribution:

Inverse gamma distribution and GammaDistribution have an inverse relationship:

LevyDistribution

is a special case of inverse gamma distribution:

Inverse gamma distribution is a special case of type 5 PearsonDistribution:

Generalized inverse gamma distribution simplifies to inverse gamma distribution:

Possible Issues (2)

InverseGammaDistribution is not defined when either

or

is not a positive real number:

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