This paper proposes a Markov Decision Process (MDP) approach to compute theoptimal on-line speed scaling policy that minimizes the energy consumption of a single processorexecuting 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 lifeapplications, it is common at release time to know only the Worst-Case Execution Time of a job,and theactualexecution time of this job is only discovered when it finishes. Choosing the processorspeed purely in function of the Worst-Case Execution Time is sub-optimal. When the probabilitydistribution of the actual execution time is known, it is possible to exploit this knowledge tochoose a lower processor speed so as to minimize the expected energy consumption (while stillguaranteeing that all jobs meet their deadline). Our MDP solution solves this problem optimallywith discrete processor speeds. Compared with approaches from the literature, the gain offeredby the new policy ranges from a few percent when the variability of job characteristics is small, tomore than50%when the job execution time distributions are far from their worst case.