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124 lines
4.0 KiB
Plaintext
124 lines
4.0 KiB
Plaintext
[section:normal_dist Normal (Gaussian) Distribution]
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``#include <boost/math/distributions/normal.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 normal_distribution;
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typedef normal_distribution<> normal;
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template <class RealType, class ``__Policy``>
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class normal_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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BOOST_MATH_GPU_ENABLED normal_distribution(RealType mean = 0, RealType sd = 1);
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// Accessors:
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BOOST_MATH_GPU_ENABLED RealType mean()const; // location.
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BOOST_MATH_GPU_ENABLED RealType standard_deviation()const; // scale.
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// Synonyms, provided to allow generic use of find_location and find_scale.
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BOOST_MATH_GPU_ENABLED RealType location()const;
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BOOST_MATH_GPU_ENABLED RealType scale()const;
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};
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}} // namespaces
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The normal distribution is probably the most well known statistical
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distribution: it is also known as the Gaussian Distribution.
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A normal distribution with mean zero and standard deviation one
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is known as the ['Standard Normal Distribution].
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Given mean [mu] and standard deviation [sigma] it has the PDF:
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[equation normal_ref1]
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The variation the PDF with its parameters is illustrated
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in the following graph:
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[graph normal_pdf]
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The cumulative distribution function is given by
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[equation normal_cdf]
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and illustrated by this graph
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[graph normal_cdf]
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[h4 Member Functions]
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BOOST_MATH_GPU_ENABLED normal_distribution(RealType mean = 0, RealType sd = 1);
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Constructs a normal distribution with mean /mean/ and
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standard deviation /sd/.
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Requires /sd/ > 0, otherwise __domain_error is called.
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BOOST_MATH_GPU_ENABLED RealType mean()const;
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BOOST_MATH_GPU_ENABLED RealType location()const;
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both return the /mean/ of this distribution.
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BOOST_MATH_GPU_ENABLED RealType standard_deviation()const;
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BOOST_MATH_GPU_ENABLED RealType scale()const;
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both return the /standard deviation/ of this distribution.
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(Redundant location and scale function are provided to match other similar distributions,
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allowing the functions find_location and find_scale to be used generically).
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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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For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
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be run on both host and device.
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The domain of the random variable is \[-[max_value], +[min_value]\].
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However, the pdf of +[infin] and -[infin] = 0 is also supported,
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and cdf at -[infin] = 0, cdf at +[infin] = 1,
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and complement cdf -[infin] = 1 and +[infin] = 0,
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if RealType permits.
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[h4 Accuracy]
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The normal distribution is implemented in terms of 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 mean of the distribution,
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and /s/ is its standard deviation.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = e[super -(x-m)[super 2]\/(2s[super 2])] \/ (s * sqrt(2*pi)) ]]
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[[logpdf][log(pdf) = -log(s) - log(2*[pi])/2 - (x-mean)[super 2]/(2*s[super 2]) ]]
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[[cdf][Using the relation: p = 0.5 * __erfc(-(x-m)/(s*sqrt(2))) ]]
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[[cdf complement][Using the relation: q = 0.5 * __erfc((x-m)/(s*sqrt(2))) ]]
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[[quantile][Using the relation: x = m - s * sqrt(2) * __erfc_inv(2*p)]]
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[[quantile from the complement][Using the relation: x = m + s * sqrt(2) * __erfc_inv(2*p)]]
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[[mean and standard deviation][The same as `dist.mean()` and `dist.standard_deviation()`]]
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[[mode][The same as the mean.]]
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[[median][The same as the mean.]]
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[[skewness][0]]
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[[kurtosis][3]]
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[[kurtosis excess][0]]
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]
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[endsect] [/section:normal_dist Normal]
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[/ normal.qbk
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Copyright 2006, 2007, 2012 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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