Logistic Distribution

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##### Logistic Distribution

    #include <boost/math/distributions/logistic.hpp>

    namespace boost{ namespace math{

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

    template <class RealType, class Policy>
    class logistic_distribution
    {
    public:
       typedef RealType value_type;
       typedef Policy   policy_type;
       // Construct:
       logistic_distribution(RealType location = 0, RealType scale = 1);
       // Accessors:
       RealType location()const; // location.
       RealType scale()const; // scale.

    };

    typedef logistic_distribution<> logistic;

    }} // namespaces

The logistic distribution is a continous probability distribution. It has two parameters - location and scale. The cumulative distribution function of the logistic distribution appears in logistic regression and feedforward neural networks. Among other applications, United State Chess Federation and FIDE use it to calculate chess ratings.

The following graph shows how the distribution changes as the parameters change:

##### Member Functions

    logistic_distribution(RealType u = 0, RealType s = 1);

Constructs a logistic distribution with location u and scale s.

Requires `scale > 0`, otherwise a domain_error is raised.

    RealType location()const;

Returns the location of this distribution.

    RealType scale()const;

Returns the scale of this distribution.

##### 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 and cdf support inputs of +∞ and -∞ as special cases if RealType permits.

At `p=1` and `p=0`, the quantile function returns the result of +overflow_error and -overflow_error, while the complement quantile function returns the result of -overflow_error and +overflow_error respectively.

##### Accuracy

The logistic distribution is implemented in terms of the `std::exp` and the `std::log` functions, so its accuracy is related to the accurate implementations of those functions on a given platform. When calculating the quantile with a non-zero position parameter catastrophic cancellation errors can occur: in such cases, only a low absolute error can be guarenteed.

##### Implementation

Function | Implementation Notes
--- | ---
pdf | Using the relation: pdf = e-(x-u)/s / (s*(1+e-(x-u)/s)2)
cdf | Using the relation: p = 1/(1+e-(x-u)/s)
cdf complement | Using the relation: q = 1/(1+e(x-u)/s)
quantile | Using the relation: x = u - s*log(1/p-1)
quantile from the complement | Using the relation: x = u + s*log(p/1-p)
mean | u
mode | The same as the mean.
skewness | 0
kurtosis excess | 6/5
variance | (π*s)2 / 3