A general procedure for obtaining exact equations between statistical mechanics expectation values is suggested. The procedure is illustrated for quantum and classical systems in a generalized ensemble. Detailed applications are given for the Stockmayer potential in classical statistical thermodynamics. Application to site-site interactions is also discussed. The resulting equations can be used to detect inconsistencies in approximate theories. This is illustrated for the Percus-Yevick equations for hard chain molecules. For a complete description of equilibrium properties, additional information is needed to supplement the derived equations.