In many optimization problems the objective function may depend on a random set of coefficients that have some known distribution. For example, a profit function may depend on certain market conditions which can at best be estimated by a set of random variables with some given distribution. An important question to be answered in such problems is: What is the expected value of the maximum profit? On the answer to this question may hinge certain very important decisions that an organization may have to make. It may also provide good estimates of future profits. While the problem of determining the expected value of the maximum may be very difficult at times, a number of related problems, some quite important in their own right, may be considerably easier to solve and also may provide bounds for the expected value of the maximum. In this work, one upper and three lower bounds are given in terms of the solutions of related problems. In addition a convexity property for a class of parametric nonlinear programming problems is obtained. This property is also used in deriving some of the bounds.