Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a 1< a 2< a 3<…. In particular, a i could be equal to i for all i. Let X 1 n ≦ X 2 n ≦⋯≦ X nn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦ k≦ n), independence of the event { X kn = X 1 n } and X 1 n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{ X kn = X 1 n | X 1 n = a i } is equivalent to X having the IFR (DFR) property. Let a i = i and G(i) = P(X≧i), i = 1, 2, …. We prove that the independence of { X 2 n − X 1 n ∈ B} and X 1 n for all i is equivalent to X being geometric, where B = { m} ( B = { m, m+1,…}), provided G( i) = q i−1 , 1≦ i≦ m+2 (1≦ i≦ m+1), where 0< q<1.