Summary: In this paper we present some characteristic properties of the Pareto, Lomax, exponential and beta distributions in terms of their \(r\)-th partial moments: \(\alpha_r (t) = \mathrm{E}[(\max\{X - t, 0\})^r]\). Thus, for example, the relation \(\alpha_r (s) \alpha_r (t) = \alpha_r (1) \alpha_r (st)\) for \(s, t > 1\) and for a positive integer \(r\) is the characteristic property of the Pareto distribution.