Application of the Box-Muller Transformation in Generating Normally Distributed Random Variables: A Numerical Approach

The Box-Muller transformation has traditionally become one of the techniques in generating normally distributed random variables based on uniformly distributed variables. It is so important in simulations, statistical analyses, and stochastic modeling, since normal distributions are very often inevitable there. Normally distributed variables were generated, their properties were discussed for five examples in numerical terms, while presenting the Box-Muller transformation. Examples: First - Uniform random variables to normally distributed. Second - Using the transformation to simulate financial returns. Third: Checking how this transformation could be used to generate data in machine learning algorithms. Each of these examples serves to demonstrate how the Box-Muller method allows one to attain, with great efficiency, normal distributions in various domains. In turn, results against theoretical normal distributions validate the output of the transformation. The excellent performance of the Box-Muller transformation in generating Gaussian random variables can be obtained from the above results, and these are quite versatile in application. This proposal underlines the simplicity, efficiency, and reliability of the transformation as an indispensable tool for the researcher and practitioner.