In this paper we discuss a stochastic scheduling problem with impatience to the beginning of service. The impatience of a job can be seen as a due date and a job is considered to be on time if it begins to be processed before its due-date. Processing times and due dates are random variables. Jobs are processed on a single machine with the objective to minimize the expected weighted number of tardy jobs in the class of static list scheduling policies. We derive optimal schedules when processing times and due dates follow different probability distributions. We also study the relation between the problem with impatience to the beginning of service and the problem with impatience to the end of service. Finally, we provide some additional results for the problem with impatience to the end of service.