Probabilities, intervals, what next? Optimization problems related to extension of interval computations to situations with partial information about probabilities

When we have only interval ranges [(x) under bar, (x) over bar (i)] of sample values x(1),...,x(n), what is the interval [V, (V) over bar] of possible values for the variance V of these values? We show that the problem of computing the upper bound (V) over bar is NP-hard. We provide a feasible ( quadratic time) algorithm for computing the exact lower bound V on the variance of interval data. We also provide feasible algorithms that computes V under reasonable easily veri. able conditions, in particular, in case interval uncertainty is introduced to maintain privacy in a statistical database. We also extend the main formulas of interval arithmetic for different arithmetic operations x(1) op x(2) to the case when, for each input x(i), in addition to the interval x(i) = [(x) under bar (i), (x) over bar (i)] of possible values, we also know its mean E-i (or an interval E-i of possible values of the mean), and we want to find the corresponding bounds for y = x(1) op x(2) and its mean. In this case, we are interested not only in the bounds for y, but also in the bounds for the mean of y. We formulate and solve the corresponding optimization problems, and describe remaining open problems.