Simulation of correlated discrete variables with specified marginals and covariances have important applications. For example, there is a need in neuroscience research to simulate discrete neural spike train data across brain regions. In this talk, we present a new graphical approach for simulating such random variables using an efficient one-pass algorithm, where the random sample is drawn for each variable in one iteration. We also give the conditions for compatibility of the marginal probabilities and covariances. This one-pass algorithm also leads to the construction of a family of Markov random fields on a directed acyclic graph with conditional and joint field distributions. A necessary and sufficient condition that guarantees the permutation property of the derived random field is studied.