We study the problem of dividing a prize with random payoffs among a group of agents. The agents may have varying levels of entitlements, or shares, to the prize and only the means of their payoff distributions are known. In the design of allocation rules, our central objective is to guarantee, in ex ante terms, each agent at least his share of the expected payoff from the prize, a requirement known as fair share. We characterize the family of all rules satisfying fair share by means of bistochastic matrices that determine a lower bound on each agent's probability assignment. This result allows us to define the top-heavy rules and show that in terms of utilitarian social welfare, they dominate all rules satisfying fair share. Thus, the top-heavy rules are uniquely efficient subject to fair share. Our model and findings generalize those of Bogomolnaia, Moulin, and Sandomirskiy (2022, Management Science).