Parity-Based Statistics and Combinatorial Identities

Notable discrete probability laws appear as posterior distributions in the estimation of the common mean with heterogeneous variances. These probabilities, which are defined by an arbitrary set of distinct real numbers, also arise in seemingly unrelated areas of polynomial approximation and statistical physics. The corresponding orthogonal polynomials possess an interesting self-duality property, which invites the study of statistical distributions based on data parity. These distributions provide novel, intriguing formulas for the classical hypergeometric function along with several combinatorial identities.