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Standard library header <random> (C++11)

This header is part of the pseudo-random number generation library.

Includes
(C++11)
 std::initializer_list class template

Concepts
Uniform random bit generator requirements
(C++20)
 specifies that a type qualifies as a uniform random bit generator
(concept)

Classes
Random number engines
(C++11)
 implements linear congruential algorithm
(class template)
(C++11)
 implements Mersenne twister algorithm
(class template)
(C++11)
 implements a subtract-with-carry (lagged Fibonacci) algorithm
(class template)
Random number engine adaptors
(C++11)
 discards some output of a random number engine
(class template)
(C++11)
 packs the output of a random number engine into blocks of a specified number of bits
(class template)
(C++11)
 delivers the output of a random number engine in a different order
(class template)
Predefined generators
minstd_rand0 (C++11) std::linear_congruential_engine<std::uint_fast32_t, 16807, 0, 2147483647>

Discovered in 1969 by Lewis, Goodman and Miller, adopted as "Minimal standard" in 1988 by Park and Miller.
minstd_rand (C++11) std::linear_congruential_engine<std::uint_fast32_t, 48271, 0, 2147483647>

Newer "Minimum standard", recommended by Park, Miller, and Stockmeyer in 1993.
mt19937(C++11)

std::mersenne_twister_engine<std::uint_fast32_t, 32, 624, 397, 31,
0x9908b0df, 11,
0xffffffff, 7,
0x9d2c5680, 15,
0xefc60000, 18, 1812433253>
32-bit Mersenne Twister by Matsumoto and Nishimura, 1998.
mt19937_64(C++11)

std::mersenne_twister_engine<std::uint_fast64_t, 64, 312, 156, 31,
0xb5026f5aa96619e9, 29,
0x5555555555555555, 17,
0x71d67fffeda60000, 37,
0xfff7eee000000000, 43, 6364136223846793005>
64-bit Mersenne Twister by Matsumoto and Nishimura, 2000.
ranlux24_base (C++11) std::subtract_with_carry_engine<std::uint_fast32_t, 24, 10, 24>
ranlux48_base (C++11) std::subtract_with_carry_engine<std::uint_fast64_t, 48, 5, 12>
ranlux24(C++11) std::discard_block_engine<std::ranlux24_base, 223, 23>

24-bit RANLUX generator by Martin Lüscher and Fred James, 1994.
ranlux48(C++11) std::discard_block_engine<std::ranlux48_base, 389, 11>

48-bit RANLUX generator by Martin Lüscher and Fred James, 1994.
knuth_b (C++11) std::shuffle_order_engine<std::minstd_rand0, 256>
default_random_engine(C++11) implementation-defined
Non-deterministic random numbers
(C++11)
 non-deterministic random number generator using hardware entropy source
(class)
Uniform distributions
(C++11)
 produces integer values evenly distributed across a range
(class template)
(C++11)
 produces real values evenly distributed across a range
(class template)
Bernoulli distributions
(C++11)
 produces bool values on a Bernoulli distribution.
(class)
(C++11)
 produces integer values on a binomial distribution.
(class template)
(C++11)
 produces integer values on a negative binomial distribution.
(class template)
(C++11)
 produces integer values on a geometric distribution.
(class template)
Poisson distributions
(C++11)
 produces integer values on a Poisson distribution
(class template)
(C++11)
 produces real values on an exponential distribution.
(class template)
(C++11)
 produces real values on a gamma distribution.
(class template)
(C++11)
 produces real values on a Weibull distribution.
(class template)
(C++11)
 produces real values on an extreme value distribution.
(class template)
Normal distributions
(C++11)
 produces real values on a standard normal (Gaussian) distribution.
(class template)
(C++11)
 produces real values on a lognormal distribution.
(class template)
(C++11)
 produces real values on a chi-squared distribution.
(class template)
(C++11)
 produces real values on a Cauchy distribution.
(class template)
(C++11)
 produces real values on a Fisher's F-distribution.
(class template)
(C++11)
 produces real values on a Student's t-distribution.
(class template)
Sampling distributions
(C++11)
 produces random integers on a discrete distribution.
(class template)
(C++11)
 produces real values distributed on constant subintervals.
(class template)
(C++11)
 produces real values distributed on defined subintervals.
(class template)
Utilities
(C++11)
 general-purpose bias-eliminating scrambled seed sequence generator
(class)

Functions
(C++11)
 evenly distributes real values of given precision across [​0​, 1)
(function template)

Synopsis

#include <initializer_list>
 
namespace std {
  // uniform random bit generator requirements
  template<class G>
    concept uniform_random_bit_generator = /* see description */;
 
  // class template linear_congruential_engine
  template<class UIntType, UIntType a, UIntType c, UIntType m>
    class linear_congruential_engine;
 
  // class template mersenne_twister_engine
  template<class UIntType, size_t w, size_t n, size_t m, size_t r,
           UIntType a, size_t u, UIntType d, size_t s,
           UIntType b, size_t t,
           UIntType c, size_t l, UIntType f>
    class mersenne_twister_engine;
 
  // class template subtract_with_carry_engine
  template<class UIntType, size_t w, size_t s, size_t r>
    class subtract_with_carry_engine;
 
  // class template discard_block_engine
  template<class Engine, size_t p, size_t r>
    class discard_block_engine;
 
  // class template independent_bits_engine
  template<class Engine, size_t w, class UIntType>
    class independent_bits_engine;
 
  // class template shuffle_order_engine
  template<class Engine, size_t k>
    class shuffle_order_engine;
 
  // engines and engine adaptors with predefined parameters
  using minstd_rand0  = /* see description */;
  using minstd_rand   = /* see description */;
  using mt19937       = /* see description */;
  using mt19937_64    = /* see description */;
  using ranlux24_base = /* see description */;
  using ranlux48_base = /* see description */;
  using ranlux24      = /* see description */;
  using ranlux48      = /* see description */;
  using knuth_b       = /* see description */;
 
  using default_random_engine = /* see description */;
 
  // class random_device
  class random_device;
 
  // class seed_seq
  class seed_seq;
 
  // function template generate_canonical
  template<class RealType, size_t bits, class URBG>
    RealType generate_canonical(URBG& g);
 
  // class template uniform_int_distribution
  template<class IntType = int>
    class uniform_int_distribution;
 
  // class template uniform_real_distribution
  template<class RealType = double>
    class uniform_real_distribution;
 
  // class bernoulli_distribution
  class bernoulli_distribution;
 
  // class template binomial_distribution
  template<class IntType = int>
    class binomial_distribution;
 
  // class template geometric_distribution
  template<class IntType = int>
    class geometric_distribution;
 
  // class template negative_binomial_distribution
  template<class IntType = int>
    class negative_binomial_distribution;
 
  // class template poisson_distribution
  template<class IntType = int>
    class poisson_distribution;
 
  // class template exponential_distribution
  template<class RealType = double>
    class exponential_distribution;
 
  // class template gamma_distribution
  template<class RealType = double>
    class gamma_distribution;
 
  // class template weibull_distribution
  template<class RealType = double>
    class weibull_distribution;
 
  // class template extreme_value_distribution
  template<class RealType = double>
    class extreme_value_distribution;
 
  // class template normal_distribution
  template<class RealType = double>
    class normal_distribution;
 
  // class template lognormal_distribution
  template<class RealType = double>
    class lognormal_distribution;
 
  // class template chi_squared_distribution
  template<class RealType = double>
    class chi_squared_distribution;
 
  // class template cauchy_distribution
  template<class RealType = double>
    class cauchy_distribution;
 
  // class template fisher_f_distribution
  template<class RealType = double>
    class fisher_f_distribution;
 
  // class template student_t_distribution
  template<class RealType = double>
    class student_t_distribution;
 
  // class template discrete_distribution
  template<class IntType = int>
    class discrete_distribution;
 
  // class template piecewise_constant_distribution
  template<class RealType = double>
    class piecewise_constant_distribution;
 
  // class template piecewise_linear_distribution
  template<class RealType = double>
    class piecewise_linear_distribution;
}

Concept uniform_random_bit_generator 

namespace std {
  template<class G>
    concept uniform_random_bit_generator =
      invocable<G&> && unsigned_integral<invoke_result_t<G&>> &&
      requires {
        { G::min() } -> same_as<invoke_result_t<G&>>;
        { G::max() } -> same_as<invoke_result_t<G&>>;
        requires bool_constant<(G::min() < G::max())>::value;
      };
}

Class template std::linear_congruential_engine 

namespace std {
  template<class UIntType, UIntType a, UIntType c, UIntType m>
    class linear_congruential_engine {
    public:
      // types
      using result_type = UIntType;
 
      // engine characteristics
      static constexpr result_type multiplier = a;
      static constexpr result_type increment = c;
      static constexpr result_type modulus = m;
      static constexpr result_type min() { return c == 0u ? 1u: 0u; }
      static constexpr result_type max() { return m - 1u; }
      static constexpr result_type default_seed = 1u;
 
      // constructors and seeding functions
      linear_congruential_engine() : linear_congruential_engine(default_seed) {}
      explicit linear_congruential_engine(result_type s);
      template<class Sseq> explicit linear_congruential_engine(Sseq& q);
      void seed(result_type s = default_seed);
      template<class Sseq> void seed(Sseq& q);
 
      // equality operators
      friend bool operator==(const linear_congruential_engine& x,
                             const linear_congruential_engine& y);
 
      // generating functions
      result_type operator()();
      void discard(unsigned long long z);
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const linear_congruential_engine& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, linear_congruential_engine& x);
    };
}

Class template std::mersenne_twister_engine 

namespace std {
  template<class UIntType, size_t w, size_t n, size_t m, size_t r,
           UIntType a, size_t u, UIntType d, size_t s,
           UIntType b, size_t t,
           UIntType c, size_t l, UIntType f>
    class mersenne_twister_engine {
    public:
      // types
      using result_type = UIntType;
 
      // engine characteristics
      static constexpr size_t word_size = w;
      static constexpr size_t state_size = n;
      static constexpr size_t shift_size = m;
      static constexpr size_t mask_bits = r;
      static constexpr UIntType xor_mask = a;
      static constexpr size_t tempering_u = u;
      static constexpr UIntType tempering_d = d;
      static constexpr size_t tempering_s = s;
      static constexpr UIntType tempering_b = b;
      static constexpr size_t tempering_t = t;
      static constexpr UIntType tempering_c = c;
      static constexpr size_t tempering_l = l;
      static constexpr UIntType initialization_multiplier = f;
      static constexpr result_type min() { return 0; }
      static constexpr result_type max() { return /* pow(2, w) - 1 */; }
      static constexpr result_type default_seed = 5489u;
 
      // constructors and seeding functions
      mersenne_twister_engine() : mersenne_twister_engine(default_seed) {}
      explicit mersenne_twister_engine(result_type value);
      template<class Sseq> explicit mersenne_twister_engine(Sseq& q);
      void seed(result_type value = default_seed);
      template<class Sseq> void seed(Sseq& q);
 
      // equality operators
      friend bool operator==(const mersenne_twister_engine& x, const mersenne_twister_engine& y);
 
      // generating functions
      result_type operator()();
      void discard(unsigned long long z);
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const mersenne_twister_engine& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, mersenne_twister_engine& x);
    };
}

Class template std::subtract_with_carry_engine 

namespace std {
  template<class UIntType, size_t w, size_t s, size_t r>
    class subtract_with_carry_engine {
    public:
      // types
      using result_type = UIntType;
 
      // engine characteristics
      static constexpr size_t word_size = w;
      static constexpr size_t short_lag = s;
      static constexpr size_t long_lag = r;
      static constexpr result_type min() { return 0; }
      static constexpr result_type max() { return /* pow(2, w) - 1 */; }
      static constexpr result_type default_seed = 19780503u;
 
      // constructors and seeding functions
      subtract_with_carry_engine() : subtract_with_carry_engine(default_seed) {}
      explicit subtract_with_carry_engine(result_type value);
      template<class Sseq> explicit subtract_with_carry_engine(Sseq& q);
      void seed(result_type value = default_seed);
      template<class Sseq> void seed(Sseq& q);
 
      // equality operators
      friend bool operator==(const subtract_with_carry_engine& x,
                             const subtract_with_carry_engine& y);
 
      // generating functions
      result_type operator()();
      void discard(unsigned long long z);
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const subtract_with_carry_engine& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, subtract_with_carry_engine& x);
    };
}

Class template std::discard_block_engine 

namespace std {
  template<class Engine, size_t p, size_t r>
    class discard_block_engine {
    public:
      // types
      using result_type = typename Engine::result_type;
 
      // engine characteristics
      static constexpr size_t block_size = p;
      static constexpr size_t used_block = r;
      static constexpr result_type min() { return Engine::min(); }
      static constexpr result_type max() { return Engine::max(); }
 
      // constructors and seeding functions
      discard_block_engine();
      explicit discard_block_engine(const Engine& e);
      explicit discard_block_engine(Engine&& e);
      explicit discard_block_engine(result_type s);
      template<class Sseq> explicit discard_block_engine(Sseq& q);
      void seed();
      void seed(result_type s);
      template<class Sseq> void seed(Sseq& q);
 
      // equality operators
      friend bool operator==(const discard_block_engine& x, const discard_block_engine& y);
 
      // generating functions
      result_type operator()();
      void discard(unsigned long long z);
 
      // property functions
      const Engine& base() const noexcept { return e; };
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const discard_block_engine& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, discard_block_engine& x);
 
    private:
      Engine e;   // exposition only
      int n;      // exposition only
    };
}

Class template std::independent_bits_engine 

namespace std {
  template<class Engine, size_t w, class UIntType>
    class independent_bits_engine {
    public:
      // types
      using result_type = UIntType;
 
      // engine characteristics
      static constexpr result_type min() { return 0; }
      static constexpr result_type max() { return /* pow(2, w) - 1 */; }
 
      // constructors and seeding functions
      independent_bits_engine();
      explicit independent_bits_engine(const Engine& e);
      explicit independent_bits_engine(Engine&& e);
      explicit independent_bits_engine(result_type s);
      template<class Sseq> explicit independent_bits_engine(Sseq& q);
      void seed();
      void seed(result_type s);
      template<class Sseq> void seed(Sseq& q);
 
      // equality operators
      friend bool operator==(const independent_bits_engine& x, const independent_bits_engine& y);
 
      // generating functions
      result_type operator()();
      void discard(unsigned long long z);
 
      // property functions
      const Engine& base() const noexcept { return e; };
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const independent_bits_engine& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, independent_bits_engine& x);
 
    private:
      Engine e;   // exposition only
    };
}

Class template std::shuffle_order_engine 

namespace std {
  template<class Engine, size_t k>
    class shuffle_order_engine {
    public:
      // types
      using result_type = typename Engine::result_type;
 
      // engine characteristics
      static constexpr size_t table_size = k;
      static constexpr result_type min() { return Engine::min(); }
      static constexpr result_type max() { return Engine::max(); }
 
      // constructors and seeding functions
      shuffle_order_engine();
      explicit shuffle_order_engine(const Engine& e);
      explicit shuffle_order_engine(Engine&& e);
      explicit shuffle_order_engine(result_type s);
      template<class Sseq> explicit shuffle_order_engine(Sseq& q);
      void seed();
      void seed(result_type s);
      template<class Sseq> void seed(Sseq& q);
 
      // equality operators
      friend bool operator==(const shuffle_order_engine& x, const shuffle_order_engine& y);
 
      // generating functions
      result_type operator()();
      void discard(unsigned long long z);
 
      // property functions
      const Engine& base() const noexcept { return e; };
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const shuffle_order_engine& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, shuffle_order_engine& x);
 
    private:
      Engine e;           // exposition only
      result_type V[k];   // exposition only
      result_type Y;      // exposition only
    };
}

Engines and engine adaptors with predefined parameters

namespace std {
  using minstd_rand0 =
        linear_congruential_engine<uint_fast32_t, 16'807, 0, 2'147'483'647>;
  using minstd_rand =
        linear_congruential_engine<uint_fast32_t, 48'271, 0, 2'147'483'647>;
  using mt19937 =
        mersenne_twister_engine<uint_fast32_t, 32, 624, 397, 31,
         0x9908'b0df, 11, 0xffff'ffff, 7, 0x9d2c'5680, 15, 0xefc6'0000, 18, 1'812'433'253>; //'
  using mt19937_64 =
        mersenne_twister_engine<uint_fast64_t, 64, 312, 156, 31,
         0xb502'6f5a'a966'19e9, 29, 0x5555'5555'5555'5555, 17,
         0x71d6'7fff'eda6'0000, 37, 0xfff7'eee0'0000'0000, 43, 6'364'136'223'846'793'005>; //'
  using ranlux24_base =
        subtract_with_carry_engine<uint_fast32_t, 24, 10, 24>;
  using ranlux48_base =
        subtract_with_carry_engine<uint_fast64_t, 48, 5, 12>;
  using ranlux24 = discard_block_engine<ranlux24_base, 223, 23>;
  using ranlux48 = discard_block_engine<ranlux48_base, 389, 11>;
  using knuth_b = shuffle_order_engine<minstd_rand0,256>;
  using default_random_engine = /* implementation-defined */;
}

Class std::random_device 

namespace std {
  class random_device {
  public:
    // types
    using result_type = unsigned int;
 
    // generator characteristics
    static constexpr result_type min() { return numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return numeric_limits<result_type>::max(); }
 
    // constructors
    random_device() : random_device(/* implementation-defined */) {}
    explicit random_device(const string& token);
 
    // generating functions
    result_type operator()();
 
    // property functions
    double entropy() const noexcept;
 
    // no copy functions
    random_device(const random_device&) = delete;
    void operator=(const random_device&) = delete;
  };
}

Class std::seed_seq 

namespace std {
  class seed_seq {
  public:
    // types
    using result_type = uint_least32_t;
 
    // constructors
    seed_seq() noexcept;
    template<class T>
      seed_seq(initializer_list<T> il);
    template<class InputIt>
      seed_seq(InputIt begin, InputIt end);
 
    // generating functions
    template<class RandomAccessIt>
      void generate(RandomAccessIt begin, RandomAccessIt end);
 
    // property functions
    size_t size() const noexcept;
    template<class OutputIt>
      void param(OutputIt dest) const;
 
    // no copy functions
    seed_seq(const seed_seq&) = delete;
    void operator=(const seed_seq&) = delete;
 
  private:
    vector<result_type> v;        // exposition only
  };
}

Class template std::uniform_int_distribution 

namespace std {
  template<class IntType = int>
    class uniform_int_distribution {
    public:
      // types
      using result_type = IntType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      uniform_int_distribution() : uniform_int_distribution(0) {}
      explicit uniform_int_distribution(IntType a,
                                        IntType b = numeric_limits<IntType>::max());
      explicit uniform_int_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const uniform_int_distribution& x, const uniform_int_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      result_type a() const;
      result_type b() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const uniform_int_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, uniform_int_distribution& x);
    };
}

Class template std::uniform_real_distribution 

namespace std {
  template<class RealType = double>
    class uniform_real_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      uniform_real_distribution() : uniform_real_distribution(0.0) {}
      explicit uniform_real_distribution(RealType a, RealType b = 1.0);
      explicit uniform_real_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const uniform_real_distribution& x,
                             const uniform_real_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      result_type a() const;
      result_type b() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const uniform_real_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, uniform_real_distribution& x);
    };
}

Class std::bernoulli_distribution 

namespace std {
  class bernoulli_distribution {
  public:
    // types
    using result_type = bool;
    using param_type  = /* unspecified */;
 
    // constructors and reset functions
    bernoulli_distribution() : bernoulli_distribution(0.5) {}
    explicit bernoulli_distribution(double p);
    explicit bernoulli_distribution(const param_type& parm);
    void reset();
 
    // equality operators
    friend bool operator==(const bernoulli_distribution& x, const bernoulli_distribution& y);
 
    // generating functions
    template<class URBG>
      result_type operator()(URBG& g);
    template<class URBG>
      result_type operator()(URBG& g, const param_type& parm);
 
    // property functions
    double p() const;
    param_type param() const;
    void param(const param_type& parm);
    result_type min() const;
    result_type max() const;
 
    // inserters and extractors
    template<class CharT, class Traits>
      friend basic_ostream<CharT, Traits>&
        operator<<(basic_ostream<CharT, Traits>& os, const bernoulli_distribution& x);
    template<class CharT, class Traits>
      friend basic_istream<CharT, Traits>&
        operator>>(basic_istream<CharT, Traits>& is, bernoulli_distribution& x);
  };
}

Class template std::binomial_distribution 

namespace std {
  template<class IntType = int>
    class binomial_distribution {
    public:
      // types
      using result_type = IntType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      binomial_distribution() : binomial_distribution(1) {}
      explicit binomial_distribution(IntType t, double p = 0.5);
      explicit binomial_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const binomial_distribution& x, const binomial_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      IntType t() const;
      double p() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const binomial_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, binomial_distribution& x);
    };
}

Class template std::geometric_distribution 

namespace std {
  template<class IntType = int>
    class geometric_distribution {
    public:
      // types
      using result_type = IntType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      geometric_distribution() : geometric_distribution(0.5) {}
      explicit geometric_distribution(double p);
      explicit geometric_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const geometric_distribution& x, const geometric_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      double p() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const geometric_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, geometric_distribution& x);
    };
}

Class template std::negative_binomial_distribution 

namespace std {
  template<class IntType = int>
    class negative_binomial_distribution {
    public:
      // types
      using result_type = IntType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      negative_binomial_distribution() : negative_binomial_distribution(1) {}
      explicit negative_binomial_distribution(IntType k, double p = 0.5);
      explicit negative_binomial_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const negative_binomial_distribution& x,
                             const negative_binomial_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      IntType k() const;
      double p() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const negative_binomial_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, negative_binomial_distribution& x);
    };
}

Class template std::poisson_distribution 

namespace std {
  template<class IntType = int>
    class poisson_distribution {
    public:
      // types
      using result_type = IntType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      poisson_distribution() : poisson_distribution(1.0) {}
      explicit poisson_distribution(double mean);
      explicit poisson_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const poisson_distribution& x, const poisson_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      double mean() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const poisson_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, poisson_distribution& x);
    };
}

Class template std::exponential_distribution 

namespace std {
  template<class RealType = double>
    class exponential_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      exponential_distribution() : exponential_distribution(1.0) {}
      explicit exponential_distribution(RealType lambda);
      explicit exponential_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const exponential_distribution& x, const exponential_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType lambda() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const exponential_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, exponential_distribution& x);
  };
}

Class template std::gamma_distribution 

namespace std {
  template<class RealType = double>
    class gamma_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      gamma_distribution() : gamma_distribution(1.0) {}
      explicit gamma_distribution(RealType alpha, RealType beta = 1.0);
      explicit gamma_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const gamma_distribution& x, const gamma_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType alpha() const;
      RealType beta() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const gamma_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, gamma_distribution& x);
    };
}

Class template std::weibull_distribution 

namespace std {
  template<class RealType = double>
    class weibull_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      weibull_distribution() : weibull_distribution(1.0) {}
      explicit weibull_distribution(RealType a, RealType b = 1.0);
      explicit weibull_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const weibull_distribution& x, const weibull_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType a() const;
      RealType b() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const weibull_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, weibull_distribution& x);
    };
}

Class template std::extreme_value_distribution 

namespace std {
  template<class RealType = double>
    class extreme_value_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      extreme_value_distribution() : extreme_value_distribution(0.0) {}
      explicit extreme_value_distribution(RealType a, RealType b = 1.0);
      explicit extreme_value_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const extreme_value_distribution& x,
                             const extreme_value_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType a() const;
      RealType b() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const extreme_value_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, extreme_value_distribution& x);
    };
}

Class template std::normal_distribution 

namespace std {
  template<class RealType = double>
    class normal_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      normal_distribution() : normal_distribution(0.0) {}
      explicit normal_distribution(RealType mean, RealType stddev = 1.0);
      explicit normal_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const normal_distribution& x, const normal_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType mean() const;
      RealType stddev() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const normal_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, normal_distribution& x);
    };
}

Class template std::lognormal_distribution 

namespace std {
  template<class RealType = double>
    class lognormal_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      lognormal_distribution() : lognormal_distribution(0.0) {}
      explicit lognormal_distribution(RealType m, RealType s = 1.0);
      explicit lognormal_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const lognormal_distribution& x, const lognormal_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType m() const;
      RealType s() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const lognormal_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, lognormal_distribution& x);
    };
}

Class template std::chi_squared_distribution 

namespace std {
  template<class RealType = double>
    class chi_squared_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      chi_squared_distribution() : chi_squared_distribution(1.0) {}
      explicit chi_squared_distribution(RealType n);
      explicit chi_squared_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const chi_squared_distribution& x, const chi_squared_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType n() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const chi_squared_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, chi_squared_distribution& x);
    };
}

Class template std::cauchy_distribution 

namespace std {
  template<class RealType = double>
    class cauchy_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      cauchy_distribution() : cauchy_distribution(0.0) {}
      explicit cauchy_distribution(RealType a, RealType b = 1.0);
      explicit cauchy_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const cauchy_distribution& x, const cauchy_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType a() const;
      RealType b() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const cauchy_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, cauchy_distribution& x);
    };

Class template std::fisher_f_distribution 

namespace std {
  template<class RealType = double>
    class fisher_f_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      fisher_f_distribution() : fisher_f_distribution(1.0) {}
      explicit fisher_f_distribution(RealType m, RealType n = 1.0);
      explicit fisher_f_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const fisher_f_distribution& x, const fisher_f_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType m() const;
      RealType n() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const fisher_f_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, fisher_f_distribution& x);
    };
}

Class template std::student_t_distribution 

namespace std {
  template<class RealType = double>
    class student_t_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      student_t_distribution() : student_t_distribution(1.0) {}
      explicit student_t_distribution(RealType n);
      explicit student_t_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const student_t_distribution& x, const student_t_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      RealType n() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
       // inserters and extractors
       template<class CharT, class Traits>
         friend basic_ostream<CharT, Traits>&
           operator<<(basic_ostream<CharT, Traits>& os, const student_t_distribution& x);
       template<class CharT, class Traits>
         friend basic_istream<CharT, Traits>&
           operator>>(basic_istream<CharT, Traits>& is, student_t_distribution& x);
    };
}

Class template std::discrete_distribution 

namespace std {
  template<class IntType = int>
    class discrete_distribution {
    public:
      // types
      using result_type = IntType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      discrete_distribution();
      template<class InputIt>
        discrete_distribution(InputIt firstW, InputIt lastW);
      discrete_distribution(initializer_list<double> wl);
      template<class UnaryOperation>
        discrete_distribution(size_t nw, double xmin, double xmax, UnaryOperation fw);
      explicit discrete_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const discrete_distribution& x, const discrete_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      vector<double> probabilities() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const discrete_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, discrete_distribution& x);
    };
}

Class template std::piecewise_constant_distribution 

namespace std {
   template<class RealType = double>
    class piecewise_constant_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      piecewise_constant_distribution();
      template<class InputItB, class InputItW>
        piecewise_constant_distribution(InputItB firstB, InputItB lastB, InputItW firstW);
      template<class UnaryOperation>
        piecewise_constant_distribution(initializer_list<RealType> bl, UnaryOperation fw);
      template<class UnaryOperation>
        piecewise_constant_distribution(size_t nw, RealType xmin, RealType xmax,
                                        UnaryOperation fw);
      explicit piecewise_constant_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const piecewise_constant_distribution& x,
                             const piecewise_constant_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      vector<result_type> intervals() const;
      vector<result_type> densities() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const piecewise_constant_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, piecewise_constant_distribution& x);
    };
}

Class template std::piecewise_linear_distribution 

namespace std {
  template<class RealType = double>
    class piecewise_linear_distribution {
    public:
      // types
      using result_type = RealType;
      using param_type  = /* unspecified */;
 
      // constructors and reset functions
      piecewise_linear_distribution();
      template<class InputItB, class InputItW>
        piecewise_linear_distribution(InputItB firstB, InputItB lastB, InputItW firstW);
      template<class UnaryOperation>
        piecewise_linear_distribution(initializer_list<RealType> bl, UnaryOperation fw);
      template<class UnaryOperation>
        piecewise_linear_distribution(size_t nw, RealType xmin, RealType xmax,
                                      UnaryOperation fw);
      explicit piecewise_linear_distribution(const param_type& parm);
      void reset();
 
      // equality operators
      friend bool operator==(const piecewise_linear_distribution& x,
                             const piecewise_linear_distribution& y);
 
      // generating functions
      template<class URBG>
        result_type operator()(URBG& g);
      template<class URBG>
        result_type operator()(URBG& g, const param_type& parm);
 
      // property functions
      vector<result_type> intervals() const;
      vector<result_type> densities() const;
      param_type param() const;
      void param(const param_type& parm);
      result_type min() const;
      result_type max() const;
 
      // inserters and extractors
      template<class CharT, class Traits>
        friend basic_ostream<CharT, Traits>&
          operator<<(basic_ostream<CharT, Traits>& os, const piecewise_linear_distribution& x);
      template<class CharT, class Traits>
        friend basic_istream<CharT, Traits>&
          operator>>(basic_istream<CharT, Traits>& is, piecewise_linear_distribution& x);
    };
}

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