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- [section:normal_dist Normal (Gaussian) Distribution]
- ``#include <boost/math/distributions/normal.hpp>``
- namespace boost{ namespace math{
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class normal_distribution;
- typedef normal_distribution<> normal;
- template <class RealType, class ``__Policy``>
- class normal_distribution
- {
- public:
- typedef RealType value_type;
- typedef Policy policy_type;
- // Construct:
- normal_distribution(RealType mean = 0, RealType sd = 1);
- // Accessors:
- RealType mean()const; // location.
- RealType standard_deviation()const; // scale.
- // Synonyms, provided to allow generic use of find_location and find_scale.
- RealType location()const;
- RealType scale()const;
- };
- }} // namespaces
- The normal distribution is probably the most well known statistical
- distribution: it is also known as the Gaussian Distribution.
- A normal distribution with mean zero and standard deviation one
- is known as the ['Standard Normal Distribution].
- Given mean [mu] and standard deviation [sigma] it has the PDF:
- [equation normal_ref1]
- The variation the PDF with its parameters is illustrated
- in the following graph:
- [graph normal_pdf]
- The cumulative distribution function is given by
- [equation normal_cdf]
- and illustrated by this graph
- [graph normal_cdf]
- [h4 Member Functions]
- normal_distribution(RealType mean = 0, RealType sd = 1);
- Constructs a normal distribution with mean /mean/ and
- standard deviation /sd/.
- Requires /sd/ > 0, otherwise __domain_error is called.
- RealType mean()const;
- RealType location()const;
- both return the /mean/ of this distribution.
- RealType standard_deviation()const;
- RealType scale()const;
- both return the /standard deviation/ of this distribution.
- (Redundant location and scale function are provided to match other similar distributions,
- allowing the functions find_location and find_scale to be used generically).
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
- distributions are supported: __usual_accessors.
- The domain of the random variable is \[-[max_value], +[min_value]\].
- However, the pdf of +[infin] and -[infin] = 0 is also supported,
- and cdf at -[infin] = 0, cdf at +[infin] = 1,
- and complement cdf -[infin] = 1 and +[infin] = 0,
- if RealType permits.
- [h4 Accuracy]
- The normal distribution is implemented in terms of the
- [link math_toolkit.sf_erf.error_function error function],
- and as such should have very low error rates.
- [h4 Implementation]
- In the following table /m/ is the mean of the distribution,
- and /s/ is its standard deviation.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = e[super -(x-m)[super 2]\/(2s[super 2])] \/ (s * sqrt(2*pi)) ]]
- [[cdf][Using the relation: p = 0.5 * __erfc(-(x-m)/(s*sqrt(2))) ]]
- [[cdf complement][Using the relation: q = 0.5 * __erfc((x-m)/(s*sqrt(2))) ]]
- [[quantile][Using the relation: x = m - s * sqrt(2) * __erfc_inv(2*p)]]
- [[quantile from the complement][Using the relation: x = m + s * sqrt(2) * __erfc_inv(2*p)]]
- [[mean and standard deviation][The same as `dist.mean()` and `dist.standard_deviation()`]]
- [[mode][The same as the mean.]]
- [[median][The same as the mean.]]
- [[skewness][0]]
- [[kurtosis][3]]
- [[kurtosis excess][0]]
- ]
- [endsect] [/section:normal_dist Normal]
- [/ normal.qbk
- Copyright 2006, 2007, 2012 John Maddock and Paul A. Bristow.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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