NAME¶

std::cauchy_distribution - std::cauchy_distribution

Synopsis¶

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

Produces random numbers according to a Cauchy distribution (also called Lorentz
distribution):

\({\small f(x;a,b)={(b\pi{[1+{(\frac{x-a}{b})}^{2}]} })}^{-1}\)f(x; a,b) = ⎛
⎜
⎝bπ ⎡
⎢
⎣1 + ⎛
⎜
⎝

x - a
b

⎞
⎟
⎠2
⎤
⎥
⎦⎞
⎟
⎠-1

std::cauchy_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¶

a returns the distribution parameters
b (public member function)
(C++11)
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 <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 cauchy = [&gen](const float x0, const float 𝛾)
{
std::cauchy_distribution<float> d{x0 /* a */, 𝛾 /* b */};

const int norm = 1'00'00;
const float cutoff = 0.005f;

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 << "x₀ = " << x0 << ", 𝛾 = " << 𝛾 << ":\n";
draw_vbars<4,3>(bars);
for (int n : indices)
std::cout << std::setw(2) << n << " ";
std::cout << "\n\n";
};

cauchy(/* x₀ = */ -2.0f, /* 𝛾 = */ 0.50f);
cauchy(/* x₀ = */ +0.0f, /* 𝛾 = */ 1.25f);
}

Possible output:¶

x₀ = -2, 𝛾 = 0.5:
███ ┬ 0.5006
███ │
▂▂▂ ███ ▁▁▁ │
▁▁▁ ▁▁▁ ▁▁▁ ▃▃▃ ███ ███ ███ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0076
-7 -6 -5 -4 -3 -2 -1 0 1 2 3

x₀ = 0, 𝛾 = 1.25:
███ ┬ 0.2539
▅▅▅ ███ ▃▃▃ │
▁▁▁ ███ ███ ███ ▁▁▁ │
▁▁▁ ▁▁▁ ▁▁▁ ▁▁▁ ▃▃▃ ▅▅▅ ███ ███ ███ ███ ███ ▅▅▅ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0058
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 9

External links¶

Weisstein, Eric W. "Cauchy Distribution." From MathWorld — A Wolfram Web Resource.