Difference between revisions of "Log-normal distribution"

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[[Category:Distribution Functions]]
 
[[Category:Distribution Functions]]
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[[Category:Ana: Status R]]  <!-- For Lumina use, do not change -->
  
= LogNormal Distribution Function =
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= LogNormal(median, gsdev) =
  
== Declaration ==
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Generates a sample with a lognormal distribution with given '''median''' and '''gsdev''' (geometric standard deviation). The logarithm of a lognormal random variable has a normal distribution.  
LogNormal( median,gsdev,mean,stddev : optional positive ; over : ... optional atomic)
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== New to Analytica 4.0 ==
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A normal distribution is symmetric around its mean: If x := Normal(mean, sdev), then P(x <= mean - sdev) = P(x >= mean + sdev) = .15. Analogously, a lognormal distribution is ratio-symmetric around its median: If y := LogNormal(median, gsdev), then P(y <= median/gsdev) = P(y >= median*gsdev) = .15.
  
The LogNormal now supports several parametric variations, as well as an over parameter.  Specifically, you can specify a LogNormal using any of the following parameterizations:
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Lognormal actually has four parameters, '''median''', '''gsdev''' (geometric standard deviation), '''mean''', '''stddev''' (standard deviation). You can specify any two of them, which are sufficient to specify the rest. 
  LogNormal( median: med, gsdev: gs )  or just LogNormal(med,gs)
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  LogNormal(median: med, gsdev: gs)  or just LogNormal(med, gs)
  LogNormal( median: med, stddev: sd )
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  LogNormal(median: med, stddev: sd)
  LogNormal( median: med, mean: mu )
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  LogNormal(median: med, mean: mu)
  LogNormal( mean: mu, stddev: sd )
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  LogNormal(mean: mu, stddev: s)
  LogNormal( mean: mu, gsdev: gs )
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  LogNormal(mean: mu, gsdev: gs )
  LogNormal( gsdev: gs, stddev: sd )
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  LogNormal(gsdev: gs, stddev: sd)
  
If fewer than two parameters are provided, the remainder default to std-LogNormal values (e.g., such that ln(x) is standard normal).  If more than two parameters are specified, an error results.
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If you specify more than two parameters, it will give an error. If you specify no parameters, it will default to standard lognormal -- i.e. whose natural logarithm is a unit normal, mean 0 and standard deviation 1.  
  
The optional Over parameter can be used to specify dimensions over which independent sampling occurs.
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Like other distributions, you can also give one or more '''Over:''' indexes. These cause it to generate an array of independent lognormal distributions over the specified index(es). For example,
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  LogNormal(m, gsd, Over: i)
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== Syntax ==
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LogNormal(median, gsdev, mean, stddev: Optional Positive; over: ... Optional Atom)

Revision as of 01:50, 18 April 2007


LogNormal(median, gsdev)

Generates a sample with a lognormal distribution with given median and gsdev (geometric standard deviation). The logarithm of a lognormal random variable has a normal distribution.

A normal distribution is symmetric around its mean: If x := Normal(mean, sdev), then P(x <= mean - sdev) = P(x >= mean + sdev) = .15. Analogously, a lognormal distribution is ratio-symmetric around its median: If y := LogNormal(median, gsdev), then P(y <= median/gsdev) = P(y >= median*gsdev) = .15.

Lognormal actually has four parameters, median, gsdev (geometric standard deviation), mean, stddev (standard deviation). You can specify any two of them, which are sufficient to specify the rest.

LogNormal(median: med, gsdev: gs)  or just LogNormal(med, gs)
LogNormal(median: med, stddev: sd)
LogNormal(median: med, mean: mu)
LogNormal(mean: mu, stddev: s)
LogNormal(mean: mu, gsdev: gs )
LogNormal(gsdev: gs, stddev: sd)

If you specify more than two parameters, it will give an error. If you specify no parameters, it will default to standard lognormal -- i.e. whose natural logarithm is a unit normal, mean 0 and standard deviation 1.

Like other distributions, you can also give one or more Over: indexes. These cause it to generate an array of independent lognormal distributions over the specified index(es). For example,

 LogNormal(m, gsd, Over: i)

Syntax

LogNormal(median, gsdev, mean, stddev: Optional Positive; over: ... Optional Atom)
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