Implementation of a Pseudo-Random Number Generator for Normal Distribution

To implement a pseudo-random number generator for normal distribution, the following steps are required. First, we need to determine the range and quantity of numbers to be generated. Second, we must select a base random number generator algorithm such as Linear Congruential Generator (LCG) or Mersenne Twister algorithm for uniform distribution generation. The choice impacts statistical properties: LCG offers simplicity but potential periodicity issues, while Mersenne Twister provides superior period length (2^19937-1) and equidistribution properties. Third, we utilize the selected base generator to produce uniformly distributed random numbers through iterative function calls, typically implementing seed initialization and state management. Fourth, we transform these uniform random variables to standard normal distribution using methods like the Box-Muller transform (involving trigonometric functions and logarithmic operations) or Ziggurat algorithm (optimized with rejection sampling). The transformation mathematically ensures zero mean and unit variance. Finally, we scale and shift these standardized values using σ*Z+μ to generate normally distributed pseudo-random numbers with desired mean (μ) and standard deviation (σ), while validating output through statistical tests like Shapiro-Wilk or KS-test for distribution conformity.