Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic con straints involving integer valued random variables The concept of a p e cient point of a probability distribution is used to derive various equivalent problem formula tions Next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration These notions are used to derive lower and upper bounds for the optimal value of probabilistically con strained stochastic programming problems with discrete random variables The results are illustrated with numerical examples