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

# JohnsonDistribution

 JohnsonDistribution represents a bounded Johnson distribution with shape parameters , , location parameter , and scale parameter . JohnsonDistributionrepresents a semi-bounded Johnson distribution. JohnsonDistributionrepresents an unbounded Johnson distribution. JohnsonDistributionrepresents a normal Johnson distribution.
• JohnsonDistribution allows and to be any real numbers and and to be any positive real numbers.
Probability density function for bounded (SB):
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Cumulative distribution function for bounded (SB):
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Mean for bounded (SB) is available numerically:
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Variance for bounded (SB) is available numerically:
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Median for bounded (SB):
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Probability density function for bounded (SB):
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Semi-bounded (SL):
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Unbounded (SU):
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Normal (SN):
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Cumulative distribution function for bounded (SB):
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 Out[2]=
Semi-bounded (SL):
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 Out[4]=
Unbounded (SU):
 Out[5]=
 Out[6]=
Normal (SN):
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Mean for bounded (SB) is available numerically:
 Out[1]=
Semi-bounded (SL):
 Out[2]=
Unbounded (SU):
 Out[3]=
Normal (SN):
 Out[4]=

Variance for bounded (SB) is available numerically:
 Out[1]=
Semi-bounded (SL):
 Out[2]=
Unbounded (SU):
 Out[3]=
Normal (SN):
 Out[4]=

Median for bounded (SB):
 Out[1]=
Semi-bounded (SL):
 Out[2]=
Unbounded (SU):
 Out[3]=
Normal (SN):
 Out[4]=
 Scope   (7)
Generate a set of pseudorandom numbers that are bounded Johnson distributed:
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 semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Kurtosis for semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Different moments with closed forms as functions of parameters:
Bounded (SB), no closed formula for symbolic parameters, but can be evaluated numerically:
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Hazard function for bounded (SB):
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
Quantile function for bounded (SB):
Semi-bounded (SL):
Unbounded (SU):
Normal (SN):
 Applications   (1)
Johnson distribution can be used to model snowfall records:
Fit Johnson (SU) distribution into the data:
The Johnson distribution family is closed under translation and scaling by a positive factor:
Relations to other distributions:
Normal (SN) Johnson distribution is a NormalDistribution:
Normal (SN) Johnson distribution is a transformation of standard NormalDistribution:
Semi-bounded (SL) Johnson distribution as a transformation of NormalDistribution:
Assuming first:
Assuming :
Unbounded (SU) Johnson distribution as a transformation of NormalDistribution:
Bounded (SB) Johnson distribution as a transformation of a NormalDistribution:
Special case of SL Johnson distribution is a LogNormalDistribution:
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