Abstract: | Let Xi be i.i.d. random variables with finite expectations, and θi arbitrary constants, i=1,…,n. Yi=Xi+θi. The expected range of the Y's is Rn(θ1,…,θn)=E(maxYi-minYi. It is shown that the expected range is minimized if and only if θ1=?=θn. In the case where the Xi are independently and symmetrically distributed around the same constant, but not identically distributed, it is shown that θ1=?=θn are not necessarily the only (θ1,...,θn) minimizing Rn. Some lemmas which are applicable to more general problems of minimizing Rn are also given. |