The performance comparison of the optimally weighted LS estimate and the linear minimum variance estimate for a linear model with random input is presented. In this case optimally weighted LS estimate is not a linear estimate of a parameter given input and observation anymore while linear minimum variance estimate still is. Under a certain conditions on variance matrix invertibility, we show that the optimally weighted LS estimate outperforms the linear minimum variance estimates provided that they have the same a priori information on the parameter being estimated.