Normal (Gaussian) Distribution

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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
    {
    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:

##### 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.

##### Accuracy

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

##### Implementation

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

Function | Implementation Notes
--- | ---
pdf | Using the relation: pdf = e-(x-m)2/(2s2) / (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