...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#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 { 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 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
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).
All the usual nonmember 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.
Function 
Implementation Notes 


Using the relation: pdf = e^{(xm)2/(2s2)} / (s * sqrt(2*pi)) 
cdf 
Using the relation: p = 0.5 * erfc((xm)/(s*sqrt(2))) 
cdf complement 
Using the relation: q = 0.5 * erfc((xm)/(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 
mode 
The same as the mean. 
median 
The same as the mean. 
skewness 
0 
kurtosis 
3 
kurtosis excess 
0 