In this paper, a full probabilistic description of the response of a randomized SDOF system in both the time and the frequency domain is done. Considering that the damping of the structure does not simply relate to any single physical phenomenon, the sensitivity of the response to the randomness of the damping parameter is investigated. The stochastic analysis is conducted via the Probability Transformation Method therefore the first probability density function of the response is evaluated. The effect of the uncertain damping coefficient on the response of the SDOF system has been investigated through several numerical examples. From the response probability density function as well as from some response statistical indexes, it has been observed that the randomness in the damping parameter significantly affects the output functions of the system. Moreover, from the conducted analyses for different scenarios of damping mean value, it is possible to appreciate that more variability of the response occurs for the smaller damping value. This aspect has been found both, in the time and in the frequency domain analyses.