In the past few years, many energy-aware scheduling algorithms have been developed primarily using the dynamic voltage-frequency scaling (DVFS) capability which has been incorporated into recent commodity processors. However, these techniques are unsatisfied with optimizing both schedule length and energy consumption. Furthermore, most algorithms schedule tasks according to their average case execution time and not consider the task's execution cycles with probability distribution in real-world. In recognition of this, we study the problem of scheduling independent stochastic tasks with normal distribution, deadline and energy consumption budget constraints on a heterogeneous platform. We first formulate this energy-aware stochastic scheduling problem as a linear programming, which maximize the guaranteed confidence probabilities under deadline and energy consumption budget constraints. Then, we propose a heuristic energy-aware stochastic tasks scheduling algorithm (ESTS) to solve this problem, which can achieve high schedule performance for independent tasks with lower complexity. Our extensive simulation performance evaluation study, based on randomly generated stochastic applications and real-world applications, clearly demonstrate that our proposed heuristic algorithm can improve system guaranteed confidence probability and has a good trade-off between schedule length and energy consumption.