Random Number sampling method

Box-Muller Transform

Given a source of uniformly distributed random numbers, the Box-Muller Transform generates normally distributed random numbers.

Math

Suppose $x_1$ and $x_2$ are independent samples chosen from the uniform distribution on the unit interval (0, 1).

$y_1=R\cos (\Theta )=\sqrt{-2\ln x_1}\cos (2\pi x_2)$ $y_1=R\sin (\Theta )=\sqrt{-2\ln x_1}\sin (2\pi x_2).$

Then $y_1$ and $y_2$ are independent random variables with a standard normal distribution.

#include <cmath>
#include <limits>
#include <random>
#include <utility>
 
std::pair<double, double> generateGaussianNoise(double mu, double sigma)
{
    constexpr double epsilon = std::numeric_limits<double>::epsilon();
    constexpr double two_pi = 2.0 * M_PI;
 
    //initialize the random uniform number generator (runif) in a range 0 to 1
    static std::mt19937 rng(std::random_device{}()); // Standard mersenne_twister_engine seeded with rd()
    static std::uniform_real_distribution<> runif(0.0, 1.0);
 
    //create two random numbers, make sure u1 is greater than epsilon
    double u1, u2;
    do
    {
        u1 = runif(rng);
    }
    while (u1 <= epsilon);
    u2 = runif(rng);
 
    //compute z0 and z1
    auto mag = sigma * sqrt(-2.0 * log(u1));
    auto z0  = mag * cos(two_pi * u2) + mu;
    auto z1  = mag * sin(two_pi * u2) + mu;
 
    return std::make_pair(z0, z1);
}

Alternatives

• Margsalia Polar Method
• Ziggurat Algorithm