This paper proposes a new method to perform model reduction of linear time invariant (LTI) systems where parameters are random variables governed by probabilistic laws. It combines the well-known truncation balanced realization (TBR) technique together with the generalized polynomial chaos (GPC) formalism, a powerful tool for uncertainty propagation. GPC formalism is used to represent and compute a random parameter-dependent balancing transformation (RPD-BT) which puts the random LTI system in a balanced form almost surely within the probabilistic range of the uncertain parameters. Model reduction is then performed by truncating almost surely weakly controllable and observable states, yielding a random parameter dependent truncated balanced realization (RPD-TBR). The truncation error's moments are shown to be bounded by Hankel singular values' moments, which are also estimated using GPC formalism. As an illustrative example, the proposed method is applied to a simple mechanical model of a two-degrees of freedom mass-spring system with uncertain stiffness and damping. (C) 2015 Elsevier Ltd. All rights reserved.