Exact Upper Bound on the Mean of theProduct of Many Random VariablesWith Known

In practice, in addition to the intervals xi = x i ; xi ] of possible values of inputs x1; : : : ; xn, we sometimes also know their means Ei. For such cases, we provide an explicit exact (= best possible) upper bound for the mean of the product x1 : : : xn of positive values xi. 1 Formulation of the Problem Case study: practical problem from ecology. In many ecological applications (see, e.g., 6] and references therein), we have some information about the (positive) parameters x 1 ; : : : ; x n , and we are interested in the product y = x 1 : : : x n. For example, pollutant often comes from the industrial source to, say, a lake, via a chain of transitions, so the resulting concentration can be estimated as x 1 x 2 : : : x n , where x 1 is the original pollutant amount and the parameters x i (i 2) describe what portion of the pollutant goes from one link to the next one. For example, x 2 may describe the portion of the pollutant that seeps into the soil, x 3 the portion of the soil pollutant that goes from the soil into the creeks, and x 4 describes the portion of the creek's pollutant that stays in the lake. For each of these parameters, we usually know the interval x i = x i ; x i ] of possible values. In addition to the intervals x i = x i ; x i ] of possible values of x i , we often know the mean values E i. Our goal is then to nd the interval of possible values of the product y, and the bounds on the mean of this product. Since in ecological problems, we are mainly interested in the worst-case estimates , so we mainly interested in the upper bound y for the interval y and in 1