This technical note addresses an iterative learning control (ILC) design problem for discrete-time linear systems where the trial lengths could be randomly varying in the iteration domain. An ILC scheme with an iteration-average operator is introduced for tracking tasks with non-uniform trial lengths, which thus mitigates the requirement on classic ILC that all trial lengths must be identical. In addition, the identical initialization condition can be absolutely removed. The learning convergence condition of ILC in mathematical expectation is derived through rigorous analysis. As a result, the proposed ILC scheme is applicable to more practical systems. In the end, two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ILC scheme for both time-invariant and time-varying linear systems.