std::chi_squared_distribution(3)

NAME
std::chi_squared_distribution - std::chi_squared_distribution

Synopsis
   Defined in header <random>
   template< class RealType = double > (since C++11)
   class chi_squared_distribution;
   The chi_squared_distribution produces random numbers \(\small x>0\)x>0 according to
   the Chi-squared distribution:
           \({\small f(x;n) = }\frac{x^{(n/2)-1}\exp{(-x/2)} }{\Gamma{(n/2)}2^{n/2}
           }\)f(x;n) =
           x(n/2)-1
           e^-x/2
           Γ(n/2) 2n/2
   \(\small\Gamma\)Γ is the Gamma function (See also std::tgamma) and \(\small n\)n
   are the degrees of freedom (default 1).
   std::chi_squared_distribution satisfies all requirements of RandomNumberDistribution

Template parameters
   RealType - The result type generated by the generator. The effect is undefined if
              this is not one of float, double, or long double.

Member types
   Member type Definition
   result_type(C++11) RealType
   param_type(C++11) the type of the parameter set, see RandomNumberDistribution.

Member functions
   constructor constructs new distribution
   (C++11) (public member function)
   reset resets the internal state of the distribution
   (C++11) (public member function)

Generation
   operator() generates the next random number in the distribution
   (C++11) (public member function)

Characteristics
   n returns the degrees of freedom (\(\small n\)n) distribution parameter
   (C++11) (public member function)
   param gets or sets the distribution parameter object
   (C++11) (public member function)
   min returns the minimum potentially generated value
   (C++11) (public member function)
   max returns the maximum potentially generated value
   (C++11) (public member function)

Non-member functions
   operator==
   operator!= compares two distribution objects
   (C++11) (function)
   (C++11)(removed in C++20)
   operator<< performs stream input and output on pseudo-random number
   operator>> distribution
   (C++11) (function template)

Example
// Run this code
 #include <random>
 #include <iomanip>
 #include <map>
 #include <algorithm>
 #include <iostream>
 #include <vector>
 #include <cmath>
 template <int Height = 5, int BarWidth = 1, int Padding = 1, int Offset = 0, class Seq>
 void draw_vbars(Seq&& s, const bool DrawMinMax = true) {
     static_assert((Height > 0) && (BarWidth > 0) && (Padding >= 0) && (Offset >= 0));
     auto cout_n = [](auto&& v, int n = 1) { while (n-- > 0) std::cout << v; };
     const auto [min, max] = std::minmax_element(std::cbegin(s), std::cend(s));
     std::vector<std::div_t> qr;
     for (typedef decltype(*cbegin(s)) V; V e : s)
         qr.push_back(std::div(std::lerp(V(0), Height*8, (e - *min)/(*max - *min)), 8));
     for (auto h{Height}; h-- > 0; cout_n('\n')) {
         cout_n(' ', Offset);
         for (auto dv : qr) {
             const auto q{dv.quot}, r{dv.rem};
             unsigned char d[] { 0xe2, 0x96, 0x88, 0 }; // Full Block: '█'
             q < h ? d[0] = ' ', d[1] = 0 : q == h ? d[2] -= (7 - r) : 0;
             cout_n(d, BarWidth), cout_n(' ', Padding);
         }
         if (DrawMinMax && Height > 1)
             Height - 1 == h ? std::cout << "┬ " << *max:
                        h ? std::cout << "│ "
                        : std::cout << "┴ " << *min;
     }
 }
 int main() {
     std::random_device rd{};
     std::mt19937 gen{rd()};
     auto χ2 = [&gen](const float dof) {
         std::chi_squared_distribution<float> d{ dof /* n */ };
         const int norm = 1'00'00;
         const float cutoff = 0.002f;
         std::map<int, int> hist{};
         for (int n=0; n!=norm; ++n) { ++hist[std::round(d(gen))]; }
         std::vector<float> bars;
         std::vector<int> indices;
         for (auto const& [n, p] : hist) {
             if (float x = p * (1.0/norm); cutoff < x) {
                 bars.push_back(x);
                 indices.push_back(n);
             }
         }
         std::cout << "dof = " << dof << ":\n";
         draw_vbars<4,3>(bars);
         for (int n : indices) { std::cout << "" << std::setw(2) << n << " "; }
         std::cout << "\n\n";
     };
     for (float dof : {1.f, 2.f, 3.f, 4.f, 6.f, 9.f}) χ2(dof);
 }

Possible output:
 dof = 1:
 ███ ┬ 0.5271
 ███ │
 ███ ███ │
 ███ ███ ▇▇▇ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.003
  0 1 2 3 4 5 6 7 8
 dof = 2:
     ███ ┬ 0.3169
 ▆▆▆ ███ ▃▃▃ │
 ███ ███ ███ ▄▄▄ │
 ███ ███ ███ ███ ▇▇▇ ▄▄▄ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.004
  0 1 2 3 4 5 6 7 8 9 10
 dof = 3:
     ███ ▃▃▃ ┬ 0.2439
     ███ ███ ▄▄▄ │
 ▃▃▃ ███ ███ ███ ▇▇▇ ▁▁▁ │
 ███ ███ ███ ███ ███ ███ ▆▆▆ ▄▄▄ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0033
  0 1 2 3 4 5 6 7 8 9 10 11 12
 dof = 4:
     ▂▂▂ ███ ▃▃▃ ┬ 0.1864
     ███ ███ ███ ███ ▂▂▂ │
     ███ ███ ███ ███ ███ ▅▅▅ ▁▁▁ │
 ▅▅▅ ███ ███ ███ ███ ███ ███ ███ ▆▆▆ ▄▄▄ ▃▃▃ ▂▂▂ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0026
  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
 dof = 6:
             ▅▅▅ ▇▇▇ ███ ▂▂▂ ┬ 0.1351
         ▅▅▅ ███ ███ ███ ███ ▇▇▇ ▁▁▁ │
     ▁▁▁ ███ ███ ███ ███ ███ ███ ███ ▅▅▅ ▂▂▂ │
 ▁▁▁ ███ ███ ███ ███ ███ ███ ███ ███ ███ ███ ███ ▅▅▅ ▄▄▄ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0031
  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
 dof = 9:
             ▅▅▅ ▇▇▇ ███ ███ ▄▄▄ ▂▂▂ ┬ 0.1044
         ▃▃▃ ███ ███ ███ ███ ███ ███ ▅▅▅ ▁▁▁ │
     ▄▄▄ ███ ███ ███ ███ ███ ███ ███ ███ ███ ▆▆▆ ▃▃▃ │
 ▄▄▄ ███ ███ ███ ███ ███ ███ ███ ███ ███ ███ ███ ███ ███ ▆▆▆ ▄▄▄ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0034
  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

External links
     * Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web
       Resource.
     * Chi-squared distribution. From Wikipedia.