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Harmonic analysis of probability distribution functions has long served an important function in the treatment of stochastic systems. The tasks of generating moments and distributions of sums have been effectively executed in the Fourier spectrum. The properties of the Walsh-Hadamard transform of probability functions of discrete random variables is explored. Many analogies can be drawn between Fourier and Walsh analysis. In particular, it is shown that moments can be generated taking the Gibb's derivative of the Walsh spectrum and that products of Walsh spectra yield the distribution of dyadic sums. Stochastic systems with dyadic symmetry would benefit most from the properties of Walsh analysis and the computational advantages it offers. Some applications in the areas of information theory and pattern recognition are demonstrated.