Alyami, LamiaLamiaAlyamiSalinas, Hugo S.Hugo S.SalinasBakouch, Hassan S.Hassan S.BakouchKachour, MaherMaherKachourDaghestani, Amira F.Amira F.DaghestaniBapat, Sudeep R.Sudeep R.Bapat2026-07-072026-07-072025Alyami, Lamia; Salinas, Hugo S.; Bakouch, Hassan S.; Kachour, Maher; Daghestani, Amira F.; Bapat, Sudeep R. (2025). Symmetric Discrete Distributions on the Integer Line: A Versatile Family and Applications. SYMMETRY-BASEL, 17(12), 2148. https://doi.org/10.3390/sym171221482073-8994https://hdl.handle.net/20.500.12740/24666We introduce the Symmetric-Z (Sy-Z) family, a unified class of symmetric discrete distributions on the integers obtained by multiplying a three-point symmetric sign variable by an independent non-negative integer-valued magnitude. This sign-magnitude construction yields interpretable, zero-centered models with tunable mass at zero and dispersion balanced across signs, making them suitable for outcomes, such as differences of counts or discretized return increments. We derive general distributional properties, including closed-form expressions for the probability mass and cumulative distribution functions, bilateral generating functions, and even moments, and show that the tail behavior is inherited from the magnitude component. A characterization by symmetry and sign-magnitude independence is established and a distinctive operational feature is proved: for independent members of the family, the sum and the difference have the same distribution. As a central example, we study the symmetric Poisson model, providing measures of skewness, kurtosis, and entropy, together with estimation via the method of moments and maximum likelihood. Simulation studies assess finite-sample performance of the estimators, and applications to datasets from finance and education show improved goodness-of-fit relative to established integer-valued competitors. Overall, the Sy-Z framework offers a mathematically tractable and interpretable basis for modeling symmetric integer-valued outcomes across diverse domains.info:eu-repo/semantics/openAccessinteger-valued modelszero-inflated countssign-magnitude factorizationentropysymmetric Poisson distributionprobability generating functionmaximum likelihood estimationArticulohttps://doi.org/10.3390/sym17122148