This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# MinStableDistribution

 MinStableDistribution represents a generalized minimum extreme value distribution with location parameter , scale parameter , and shape parameter .
• The generalized minimum extreme value distribution gives the asymptotic distribution of the minimum value in a sample from a distribution such as the normal, Cauchy, or beta distribution.
• The probability density for value in a generalized minimum extreme value distribution is proportional to for and zero otherwise.
Probability density function:
Cumulative distribution function:
Mean and variance:
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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 Scope   (7)
Generate a set of pseudorandom numbers that follow generalized minimum extreme value 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 depends only on the shape parameter:
Limiting values:
The distribution is symmetric for:
Skewness has opposite sign to skewness of MaxStableDistribution:
Kurtosis depends only on the shape parameter:
Limiting values:
Kurtosis attains its minimum:
Kurtosis is the same as kurtosis of MaxStableDistribution:
Different moments with closed forms as functions of parameters:
Hazard function:
Quantile function:
 Applications   (4)
MinStableDistribution can be used to model the annual minimum mean daily flows. Consider the Mahanadi river and the minimum flows given in cubic meters per second:
Fit MinStableDistribution to the data:
Compare the histogram of the data to the PDF of the estimated distribution:
Find the average annual minimum mean daily flow:
Find the probability that the minimum flow is 1.5 cubic meters per second or less:
Assuming that the annual minimum flows are independent, find the probability that the minimum flow will not exceed 2 cubic meters per second for 3 consecutive years:
Simulate annual minimum mean daily flows for the next 30 years:
MinStableDistribution can be used to model yield strength:
Fit MinStableDistribution into the data:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the average yield strength:
Find the probability that the yield strength is at least 38:
Simulate the yield strength for the next 50 samples:
MinStableDistribution can be used to model size. Consider particles of fly ash with diameters given in 20 microns:
Fit the distribution into the data:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the average particle diameter:
Find the probability that the diameter is at least 200 microns:
Simulate the diameters for 100 ash particles:
MinStableDistribution can be used to model the length of Cyrtoideae radiolarians:
Fit the distribution into the data:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the average length of a cyrtoideae:
Find the probability that the length is at least 100 microns:
Simulate the lengths for 60 samples:
Parameter influence on the CDF for each :
MinStableDistribution is closed under translation and scaling by a positive factor:
CDF of MinStableDistribution solves the stability postulate equation:
Verify solution for :
Find the limit of :
Relationships to other distributions:
ExtremeValueDistribution is related to a generalized minimum extreme value distribution:
GumbelDistribution is a special case of a generalized minimum extreme value distribution:
Generalized minimum extreme value distribution is related to FrechetDistribution:
WeibullDistribution is a special case of a generalized minimum extreme value distribution:
Generalized minimum extreme value distribution is related to MaxStableDistribution:
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