Uniformly Distributed Random Numbers

The generation of uniformly distributed random numbers is one of the most fundamental tasks in computer simulation, typically achieved through pseudorandom number generators to ensure equal probability of values appearing within specified intervals. These uniformly distributed random numbers serve as the foundation for constructing more complex distributions, such as Gaussian distributions.

Methods for Generating Gaussian Distributed Random Numbers

12-Sum Method The 12-sum method is a simple and practical approximation technique based on the central limit theorem. The core implementation involves generating 12 uniformly distributed random numbers, summing them, and then subtracting 6. Since the sum of independent uniform random variables approximates a normal distribution, this method quickly produces Gaussian-distributed random numbers. Although less precise than other methods, it offers excellent computational efficiency, making it suitable for scenarios with lower accuracy requirements.

Box-Muller Transform The Box-Muller transform is a mathematically rigorous approach that directly generates two independent Gaussian random numbers from a pair of uniform random numbers. This method employs polar coordinate transformation to avoid approximation errors, making it widely used in scientific computing and financial modeling. Its advantage lies in high-precision Gaussian random number generation, though it involves slightly higher computational costs due to trigonometric and logarithmic operations.

Comparison and Selection The 12-sum method is ideal for rapidly generating large quantities of approximately normal random numbers, such as in game development or preliminary simulations. The Box-Muller transform is better suited for scenarios requiring high-precision Gaussian random numbers, like Monte Carlo simulations or rigorous scientific experiments.

Both methods rely on uniformly distributed random numbers as input, so the quality of these uniform random numbers directly impacts the final Gaussian distribution results. In practical applications, method selection requires balancing speed, accuracy, and implementation complexity.