This thesis consider the normal distribution with its important appearance in many statistical fields of applications. Some mathematical and statistical properties of the distribution have been collected and illustrated with moments and higher moments. Six related theorems have been studied in the applications of this type of distribution. The estimation manner and its properties have been illustrated throughout two methods (Moment and Maximum Likelihood methods) which are used to estimate the distribution parameters theoretically. Equality and properties of estimation have been studied throughout many well-known theorems. Five methods to approximate the cumulative distribution function have been used namely: Trapezoidal, Simpson, Gaussian, Hit or Miss and Sample mean rules. The results of these rules have been compared in its behavior and error of approximation resulted from each method. The comparison shows clearly that the last method ''Sample mean rule'' is the best method among of five methods for approximating the solution for this type of functions. In addition to that the results of each method have been represented by curves line and numerical tables for helping in reading and comparing the results of each method with each other. Finally four procedures for generating random varieties from normal distribution are discussed which are Box-Muller, Acceptance-Rejection, Central limit theorem and Tocher procedures and their efficiencies which are compared theoretically and practically by Monte Carlo simulation. The results of comparison shows that the Box-Muller procedure is the best one among three methods for this type of generation in sense of time consuming.