# std::student_t_distribution

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< cpp‎ | numeric‎ | random

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

Produces random floating-point values x, distributed according to probability density function:

  \(p(x|n) = \frac{1}{\sqrt{n\pi} } \cdot \frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})} \cdot (1+\frac{x^2}{n})^{-\frac{n+1}{2} } \)p(x|n) =

  1
  √nπ

  ·

  Γ(
  n+1
  2
  )
  Γ(
  n
  2
  )

  · ⎛
  ⎜
  ⎝1+

  x2
  n

  ⎞
  ⎟
  ⎠ -

  n+1
  2

where n is known as the number of degrees of freedom. This distribution is used when estimating the mean of an unknown normally distributed value given n+1 independent measurements, each with additive errors of unknown standard deviation, as in physical measurements. Or, alternatively, when estimating the unknown mean of a normal distribution with unknown standard deviation, given n+1 samples.

`std::student_t_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 | returns the n distribution parameter (degrees of freedom) (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()};

        std::student_t_distribution<> d{10.0f};

        const int norm = 10'000;
        const float cutoff = 0.000'3f;

        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 (const auto& [n, p] : hist)
            if (float x = p * (1.0f / norm); cutoff < x)
            {
                bars.push_back(x);
                indices.push_back(n);
            }

        for (draw_vbars<8, 5>(bars); const int n : indices)
            std::cout << " " << std::setw(2) << n << "   ";
        std::cout << '\n';
    }

Possible output:

                            █████                               ┬ 0.3753
                            █████                               │
                      ▁▁▁▁▁ █████                               │
                      █████ █████ ▆▆▆▆▆                         │
                      █████ █████ █████                         │
                      █████ █████ █████                         │
                ▄▄▄▄▄ █████ █████ █████ ▄▄▄▄▄                   │
    ▁▁▁▁▁ ▃▃▃▃▃ █████ █████ █████ █████ █████ ▃▃▃▃▃ ▁▁▁▁▁ ▁▁▁▁▁ ┴ 0.0049
     -4    -3    -2    -1     0     1     2     3     4     5

### External links

Weisstein, Eric W. "Student's t-Distribution." From MathWorld — A Wolfram Web Resource.

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