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InverseGammaDistribution

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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 .
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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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:
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 :
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:
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:
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