mirror of
https://github.com/boostorg/math.git
synced 2026-02-21 15:12:28 +00:00
e602f59025
Updated distribution docs to bring them into synch with the policy based code. Still a few "TODO" sections at present. [SVN r7545]
100 lines
2.9 KiB
Plaintext
100 lines
2.9 KiB
Plaintext
[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;
|
|
RealType standard_deviation()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][space] and standard deviation [sigma][space] it has the PDF:
|
|
|
|
[$../equations/normal_ref1.png]
|
|
|
|
The variation the PDF with its parameters is illustrated
|
|
in the following graph:
|
|
|
|
[$../graphs/normal.png]
|
|
|
|
[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;
|
|
|
|
Returns the /mean/ of this distribution.
|
|
|
|
RealType standard_deviation()const;
|
|
|
|
Returns the /standard deviation/ of this distribution.
|
|
|
|
[h4 Non-member Accessors]
|
|
|
|
All the [link math_toolkit.dist.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 \[-[infin], +[infin]\].
|
|
|
|
[h4 Accuracy]
|
|
|
|
The normal distribution is implemented in terms of the
|
|
[link math_toolkit.special.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.]]
|
|
[[skewness][0]]
|
|
[[kurtosis][3]]
|
|
[[kurtosis excess][0]]
|
|
]
|
|
|
|
[endsect][/section:normal_dist Normal]
|
|
|
|
[/ normal.qbk
|
|
Copyright 2006 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).
|
|
]
|
|
|