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Project portfolio selection is a crucial decision in many organizations, which must make informed decisions on investment, where the appropriate distribution of investment is complex, due to varying levels of risk, resource requirements, and interaction among the proposed projects. In this paper, we present a mathemat- ical model of investment for a set of projects under uncertainty of future returns. The model addresses the problem of an investor with access to a limited pool of capital, who makes decisions on investments. The problem is to decide how much to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. We discuss optimal investment decisions for the cases where the return from investment is a random variable. The single-period and multi period cases of investment decisions are considered. The paper presents closed form solutions for commonly adopted utility functions.