Let \(X\) be a discrete-time finite-state Markov process and let \(Y=h(X,Z)\), where \(Z\) is some i.i.d. sequence, and where the output \(Y\) can take a finite number of values. This paper considers the realization question: given the probabilities of all finite-length output strings, under what circumstances and how can one construct an \(X\) as above and a mapping \(h\) as above such that for some \(Z\) as above \(Y=h(X,Z)\) generates finite-length output strings with the given probabilities. The theory presented in this paper is an extension of recent theoretical developments in the positive realization problem of linear system theory. This paper is clearly written and ends with a section ``conclusion'' that stimulates further research in this area.