This article discusses the problem of utility maximization in a market with random-interval payoffs without short-selling prohibition.A novel expected utility model is given to measure an investors subjective view toward random interval wealth.Some techniques are proposed to transfer a complex programming involving interval numbers into a simple non-linear programming.Under the existence of the optimal strategy,relations between the optimal strategy and assets prices are discussed.Some properties of the maximal utility function with respect to the endowment are given.