The article by G. E. P. Box and Mervin E. Muller discusses a method for generating random normal deviates from independent random numbers, particularly useful for large-scale computer applications. The method involves using two independent random variables \( U_1 \) and \( U_2 \) from a rectangular density function on the interval (0,1). The random variables \( X_1 \) and \( X_2 \) are then defined as: \[ \begin{aligned} & X_{1}=(-2 \log _{e} U_{1})^{1 / 2} \cos 2 \pi U_{2} \\ & X_{2}=(-2 \log _{e} U_{1})^{1 / 2} \sin 2 \pi U_{2} \end{aligned} \] These variables form a pair of independent random variables from the same normal distribution with mean zero and unit variance. The justification for this method is provided through the inverse relationships and the joint density function, which confirms the desired properties. The method is motivated by the fact that the probability density of \( f(X_1, X_2) \) is constant on circles, making \( \Theta = \arctan X_2 / X_1 \) uniformly distributed, and the square of the radius vector \( r^2 = X_1^2 + X_2^2 \) has a Chi-squared distribution with two degrees of freedom. The method is noted for its reliability in the tails of the distribution and its ease of implementation, as it relies on existing library programs for trigonometric functions, logarithms, and square roots. The accuracy of the method depends on the precision of these library programs.The article by G. E. P. Box and Mervin E. Muller discusses a method for generating random normal deviates from independent random numbers, particularly useful for large-scale computer applications. The method involves using two independent random variables \( U_1 \) and \( U_2 \) from a rectangular density function on the interval (0,1). The random variables \( X_1 \) and \( X_2 \) are then defined as: \[ \begin{aligned} & X_{1}=(-2 \log _{e} U_{1})^{1 / 2} \cos 2 \pi U_{2} \\ & X_{2}=(-2 \log _{e} U_{1})^{1 / 2} \sin 2 \pi U_{2} \end{aligned} \] These variables form a pair of independent random variables from the same normal distribution with mean zero and unit variance. The justification for this method is provided through the inverse relationships and the joint density function, which confirms the desired properties. The method is motivated by the fact that the probability density of \( f(X_1, X_2) \) is constant on circles, making \( \Theta = \arctan X_2 / X_1 \) uniformly distributed, and the square of the radius vector \( r^2 = X_1^2 + X_2^2 \) has a Chi-squared distribution with two degrees of freedom. The method is noted for its reliability in the tails of the distribution and its ease of implementation, as it relies on existing library programs for trigonometric functions, logarithms, and square roots. The accuracy of the method depends on the precision of these library programs.