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## Class template non_central_chi_squared_distribution

boost::random::non_central_chi_squared_distribution

## Synopsis

    // In header: <boost/random/non_central_chi_squared_distribution.hpp>

    template<typename RealType = double>
    class non_central_chi_squared_distribution {
    public:
      // types
      typedef RealType result_type;
      typedef RealType input_type;

      // member classes/structs/unions

      class param_type {
      public:
        // types
        typedef non_central_chi_squared_distribution distribution_type;

        // construct/copy/destruct
        explicit param_type(RealType = 1, RealType = 1);

        // public member functions
        RealType k() const;
        RealType lambda() const;

        // friend functions
        template<typename CharT, typename Traits>
          friend std::basic_ostream< CharT, Traits > &
          operator<<(std::basic_ostream< CharT, Traits > &, const param_type &);
        template<typename CharT, typename Traits>
          friend std::basic_istream< CharT, Traits > &
          operator>>(std::basic_istream< CharT, Traits > &, const param_type &);
        friend bool operator==(const param_type &, const param_type &);
        friend bool operator!=(const param_type &, const param_type &);
      };

      // construct/copy/destruct
      explicit non_central_chi_squared_distribution(RealType = 1, RealType = 1);
      explicit non_central_chi_squared_distribution(const param_type &);

      // public member functions
      template<typename URNG>
        RealType operator()(URNG &, const param_type &) const;
      template<typename URNG> RealType operator()(URNG &);
      RealType k() const;
      RealType lambda() const;
      param_type param() const;
      void param(const param_type &);
      void reset();
      RealType min() const;
      RealType max() const;

      // friend functions
      template<typename CharT, typename Traits>
        friend std::basic_ostream< CharT, Traits > &
        operator<<(std::basic_ostream< CharT, Traits > &,
                   const non_central_chi_squared_distribution &);
      template<typename CharT, typename Traits>
        friend std::basic_istream< CharT, Traits > &
        operator>>(std::basic_istream< CharT, Traits > &,
                   const non_central_chi_squared_distribution &);
      friend bool operator==(const non_central_chi_squared_distribution &,
                             const non_central_chi_squared_distribution &);
      friend bool operator!=(const non_central_chi_squared_distribution &,
                             const non_central_chi_squared_distribution &);
    };

## Description

The noncentral chi-squared distribution is a real valued distribution with two parameter, `k` and `lambda`. The distribution produces values > 0.

This is the distribution of the sum of squares of k Normal distributed variates each with variance one and the sum of squares of the normal means.

The distribution function is . where is a modified Bessel function of the first kind.

The algorithm is taken from

> "Monte Carlo Methods in Financial Engineering", Paul Glasserman, 2003, XIII, 596 p, Stochastic Modelling and Applied Probability, Vol. 53, ISBN 978-0-387-21617-1, p 124, Fig. 3.5.

### `non_central_chi_squared_distribution` public construct/copy/destruct

1.     explicit non_central_chi_squared_distribution(RealType k = 1,
                                                     RealType lambda = 1);

   Construct a ``non_central_chi_squared_distribution`` object. `k` and `lambda` are the parameter of the distribution.

   Requires: k > 0 && lambda > 0

2.     explicit non_central_chi_squared_distribution(const param_type & param);

   Construct a ``non_central_chi_squared_distribution`` object from the parameter.

### `non_central_chi_squared_distribution` public member functions

1.     template<typename URNG>
         RealType operator()(URNG & eng, const param_type & param) const;

   Returns a random variate distributed according to the non central chi squared distribution specified by `param`.

2.     template<typename URNG> RealType operator()(URNG & eng);

   Returns a random variate distributed according to the non central chi squared distribution.

3.     RealType k() const;

   Returns the `k` parameter of the distribution.

4.     RealType lambda() const;

   Returns the `lambda` parameter of the distribution.

5.     param_type param() const;

   Returns the parameters of the distribution.

6.     void param(const param_type & param);

   Sets parameters of the distribution.

7.     void reset();

   Resets the distribution, so that subsequent uses does not depend on values already produced by it.

8.     RealType min() const;

   Returns the smallest value that the distribution can produce.

9.     RealType max() const;

   Returns the largest value that the distribution can produce.

### `non_central_chi_squared_distribution` friend functions

1.     template<typename CharT, typename Traits>
         friend std::basic_ostream< CharT, Traits > &
         operator<<(std::basic_ostream< CharT, Traits > & os,
                    const non_central_chi_squared_distribution & dist);

   Writes the parameters of the distribution to a `std::ostream`.

2.     template<typename CharT, typename Traits>
         friend std::basic_istream< CharT, Traits > &
         operator>>(std::basic_istream< CharT, Traits > & is,
                    const non_central_chi_squared_distribution & dist);

   reads the parameters of the distribution from a `std::istream`.

3.     friend bool operator==(const non_central_chi_squared_distribution & lhs,
                              const non_central_chi_squared_distribution & rhs);

   Returns true if two distributions have the same parameters and produce the same sequence of random numbers given equal generators.

4.     friend bool operator!=(const non_central_chi_squared_distribution & lhs,
                              const non_central_chi_squared_distribution & rhs);

   Returns true if two distributions have different parameters and/or can produce different sequences of random numbers given equal generators.