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

 TukeyLambdaDistribution[] represents Tukey's lambda distribution with shape parameter . TukeyLambdaDistributionrepresents Tukey's lambda distribution with location parameter and scale parameter . TukeyLambdaDistributionrepresents the generalized Tukey's lambda distribution with location parameter , scale parameters and , and shape parameters and .
• The quantile function for value in a Tukey lambda distribution is equal to for and , and when .
• The quantile function for value in a generalized Tukey lambda distribution is equal to for .
 Basic Examples   (15)
Quantile function for symmetric case:
Quantile function with location and scale parameters:
Quantile function for generalized Tukey's lambda distribution:
Probability density function does not have closed form but can be found numerically:
Cumulative distribution function does not have closed form but can be found numerically:
Mean and variance have closed forms:
Median:
Probability density function with location and scale:
Cumulative distribution function with location and scale:
Mean and variance with location and scale:
Median with location and scale:
Probability density function of generalized Tukey lambda distribution:
Cumulative distribution function of generalized Tukey lambda distribution:
Mean and variance of generalized Tukey lambda distribution:
Median of generalized Tukey lambda distribution:
Quantile function for symmetric case:
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Quantile function with location and scale parameters:
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Quantile function for generalized Tukey's lambda distribution:
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Probability density function does not have closed form but can be found numerically:
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Cumulative distribution function does not have closed form but can be found numerically:
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Mean and variance have closed forms:
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Median:
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Probability density function with location and scale:
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Cumulative distribution function with location and scale:
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Mean and variance with location and scale:
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Median with location and scale:
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Probability density function of generalized Tukey lambda distribution:
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Cumulative distribution function of generalized Tukey lambda distribution:
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Mean and variance of generalized Tukey lambda distribution:
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Median of generalized Tukey lambda distribution:
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 Scope   (6)
Generate a set of pseudorandom numbers that are Tukey's lambda distributed:
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 is zero for symmetric case:
With location and scale:
Kurtosis:
Location and scale does not change the kurtosis:
Kurtosis attains its minimum:
Different moments with closed forms as functions of parameters:
Moment for symmetric case:
Hazard function for symmetric case:
With location and scale:
Hazard function of generalized Tukey lambda distribution:
 Applications   (2)
Symmetric TukeyLambdaDistribution is often used to approximate other symmetric distributions. The distribution with approximates NormalDistribution:
Tukey-lambda distribution with approximately corresponds to CauchyDistribution:
Probability plot correlation coefficient plot is often used to determine best fit of symmetric TukeyLambdaDistribution to the data:
Visually confirm that is a good fit for normal samples:
Visually find best bit for standard samples from LogisticDistribution:
Parameter influence on the CDF for each :
Shifting and scaling by positive factor:
Relationships to other distributions:
For it is exactly UniformDistribution:
For it is exactly LogisticDistribution:
Tukey lambda with location and scale simplifies to one parameter case:
New in 8