The Extrema of the Expected Value of a Function of Independent Random Variables

The problem is considered of determining the least upper (or greatest lower) bound for the expected value $EK(X_1, \cdots, X_n)$ of a given function $K$ of $n$ random variables $X_1, \cdots, X_n$ under the assumption that $X_1, \cdots, X_n$ are independent and each $X_j$ has given range and satisfies $k$ conditions of the form $Eg^{(j)}_i (X_j) = c_{ij}$ for $i = 1, \cdots, k$. It is shown that under general conditions we need consider only discrete random variables $X_j$ which take on at most $k + 1$ values.