A common theme in stochastic optimization problems is that, theoretically, stochastic algorithms need to "know" relatively rich information about the underlying distributions. This is at odds with most applications, where distributions are rough predictions based on historical data. Thus, commonly, stochastic algorithms are making decisions using imperfect predicted distributions, while trying to optimize over some unknown true distributions. We consider the fundamental problem of scheduling stochastic jobs preemptively on a single machine to minimize expected mean completion time in the setting where the scheduler is only given imperfect predicted job size distributions. If the predicted distributions are perfect, then it is known that this problem can be solved optimally by the Gittins index policy. The goal of our work is to design a scheduling policy that is robust in the sense that it produces nearly optimal schedules even if there are modest discrepancies between the predicted distributions and the underlying real distributions. Our main contributions are: (1) We show that the standard Gittins index policy is not robust in this sense. If the true distributions are perturbed by even an arbitrarily small amount, then running the Gittins index policy using the perturbed distributions can lead to an unbounded increase in mean completion time. (2) We explain how to modify the Gittins index policy to make it robust, that is, to produce nearly optimal schedules, where the approximation depends on a new measure of error between the true and predicted distributions that we define. Looking forward, the approach we develop here can be applied more broadly to many other stochastic optimization problems to better understand the impact of mispredictions, and lead to the development of new algorithms that are robust against such mispredictions.