Some conditions for stochastic equality

If two random vectors are stochastically ordered and have the same marginal distributions then they must have the same joint distribution. In this article some variations of this result are obtained. For example, it is shown that, for nonnegative random vectors, only the stochastic ordering of positive linear combinations of the random variables is needed in order to obtain stochastic equality. A similar result for discrete random vectors is also given. A conjecture from the literature in this area is settled. Some applications are included.