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Normal (Gaussian) Distribution

#include <boost/math/distributions/normal.hpp>
namespace boost{ namespace math{

template <class RealType = double,
          class Policy   = policies::policy<> >
class normal_distribution;

typedef normal_distribution<> normal;

template <class RealType, class Policy>
class normal_distribution
   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 μ and standard deviation σ it has the PDF:

The variation the PDF with its parameters is illustrated in the following graph:

The cumulative distribution function is given by

and illustrated by this graph

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).

Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The domain of the random variable is [-[max_value], +[min_value]]. However, the pdf of +∞ and -∞ = 0 is also supported, and cdf at -∞ = 0, cdf at +∞ = 1, and complement cdf -∞ = 1 and +∞ = 0, if RealType permits.


The normal distribution is implemented in terms of the error function, and as such should have very low error rates.


In the following table m is the mean of the distribution, and s is its standard deviation.


Implementation Notes


Using the relation: pdf = e-(x-m)2/(2s2) / (s * sqrt(2*pi))


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)))


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()


The same as the mean.


The same as the mean.





kurtosis excess