Abstract: The theories of system identification have been highly elaborated so as to achieve the true system. This paper much discuses regarding the stochastic processes along with the divergent of whether or not the system has zero-mean under scenario of either white and coloured noise. The mathematical foundations, including mean, variance, covariance, optimal parameters, along with some modified scenarios among them, are presented in detail from various system along with some basic idea behind them. The families of auto-regressive (1) and (2) are compared both mathematical and simulation in order to obtain the best design approaching the true system, Moreover, the leastsquare algorithm is used to examine the effectiveness of some number of iteration along with ”batch” theorem.