# std::chi_squared_distribution

From cppreference.com
< cpp‎ | numeric‎ | random

Defined in header `<random>` |  |
template< class RealType = double > class chi_squared_distribution; |  | (since C++11)

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

##### Generation

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

##### Characteristics

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

### Non-member functions

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

### Example

Run this code

    #include <algorithm>
    #include <cmath>
    #include <iomanip>
    #include <iostream>
    #include <map>
    #include <random>
    #include <vector>

    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(0 < Height and 0 < BarWidth and 0 <= Padding and 0 <= Offset);

        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(*std::cbegin(s)) V; V e : s)
            qr.push_back(std::div(std::lerp(V(0), 8 * Height,
                                            (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";

            for (draw_vbars<4, 3>(bars); 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

1. | Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld — A Wolfram Web Resource.
2. | Chi-squared distribution — From Wikipedia.

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