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120 lines
4.0 KiB
Plaintext
120 lines
4.0 KiB
Plaintext
[section:lognormal_dist Log Normal Distribution]
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``#include <boost/math/distributions/lognormal.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class lognormal_distribution;
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typedef lognormal_distribution<> lognormal;
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template <class RealType, class ``__Policy``>
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class lognormal_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Construct:
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lognormal_distribution(RealType location = 0, RealType scale = 1);
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// Accessors:
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RealType location()const;
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RealType scale()const;
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};
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}} // namespaces
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The lognormal distribution is the distribution that arises
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when the logarithm of the random variable is normally distributed.
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A lognormal distribution results when the variable is the product
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of a large number of independent, identically-distributed variables.
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For location and scale parameters /m/ and /s/ it is defined by the
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probability density function:
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[equation lognormal_ref]
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The location and scale parameters are equivalent to the mean and
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standard deviation of the logarithm of the random variable.
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The following graph illustrates the effect of the location
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parameter on the PDF, note that the range of the random
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variable remains \[0,+[infin]\] irrespective of the value of the
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location parameter:
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[graph lognormal_pdf1]
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The next graph illustrates the effect of the scale parameter on the PDF:
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[graph lognormal_pdf2]
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[h4 Member Functions]
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lognormal_distribution(RealType location = 0, RealType scale = 1);
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Constructs a lognormal distribution with location /location/ and
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scale /scale/.
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The location parameter is the same as the mean of the logarithm of the
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random variate.
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The scale parameter is the same as the standard deviation of the
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logarithm of the random variate.
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Requires that the scale parameter is greater than zero, otherwise calls
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__domain_error.
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RealType location()const;
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Returns the /location/ parameter of this distribution.
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RealType scale()const;
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Returns the /scale/ parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
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distributions are supported: __usual_accessors.
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The domain of the random variable is \[0,+[infin]\].
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[h4 Accuracy]
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The lognormal distribution is implemented in terms of the
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standard library log and exp functions, plus the
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[link math_toolkit.sf_erf.error_function error function],
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and as such should have very low error rates.
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[h4 Implementation]
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In the following table /m/ is the location parameter of the distribution,
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/s/ is its scale parameter, /x/ is the random variate, /p/ is the probability
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and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = e[super -(ln(x) - m)[super 2 ] \/ 2s[super 2 ] ] \/ (x * s * sqrt(2pi)) ]]
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[[cdf][Using the relation: p = cdf(normal_distribtion<RealType>(m, s), log(x)) ]]
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[[cdf complement][Using the relation: q = cdf(complement(normal_distribtion<RealType>(m, s), log(x))) ]]
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[[quantile][Using the relation: x = exp(quantile(normal_distribtion<RealType>(m, s), p))]]
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[[quantile from the complement][Using the relation: x = exp(quantile(complement(normal_distribtion<RealType>(m, s), q)))]]
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[[mean][e[super m + s[super 2 ] / 2 ] ]]
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[[variance][(e[super s[super 2] ] - 1) * e[super 2m + s[super 2 ] ] ]]
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[[mode][e[super m - s[super 2 ] ] ]]
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[[skewness][sqrt(e[super s[super 2] ] - 1) * (2 + e[super s[super 2] ]) ]]
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[[kurtosis][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 3]]
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[[kurtosis excess][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 6 ]]
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]
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[endsect][/section:normal_dist Normal]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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