This paper proposes a Markov Decision Process (MDP) approach to compute the optimal on-line speed scaling policy that minimizes the energy consumption of a single processor executing a finite or infinite set of jobs with real-time constraints, in the non-clairvoyant case,i.e., when the actual execution time of the jobs is unknown when they are released. In real life applications, it is common at release time to know only the Worst-Case Execution Time of a job, and the actual execution time of this job is only discovered when it finishes. Choosing the processor speed purely in function of the Worst-Case Execution Time is sub-optimal. When the probability distribution of the actual execution time is known, it is possible to exploit this knowledge to choose a lower processor speed so as to minimize the expected energy consumption (while still guaranteeing that all jobs meet their deadline). Our MDP solution solves this problem optimally with discrete processor speeds. Compared with approaches from the literature, the gain offered by the new policy ranges from a few percent when the variability of job characteristics is small, tomore than 50%when the job execution time distributions are far from their worst case.