Abstract: We make a detailed study of using the order statistics to depict the geometric distribution. The following two conclusions have been demonstrated in the present paper. First, if there exists a k,\;1<k\le n, such that X_(k)-X_(1) is independent of the event \X_(1)=2\ and \X_(1)=4\, then X_1 is geometric. Second, if there exists a k,\;1<k\le n, such that X_(k)-X_(1) is independent of the event \X_(1)=3\ and \X_(1)=4\, then X_1 is geometric.