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In this paper, we define and discuss $\mathcal{R}(p,q)$- deformations ofbasic univariate discrete distributions of the probability theory. We mainlyfocus on binomial, Euler, P\'olya and inverse P\'olya distributions. We discussrelevant $\mathcal{R}(p,q)-$ deformed factorial moments of a random variable,and establish associated expressions of mean and variance. Futhermore, wederive a recursion relation for the probability distributions. Then, we applythe same approach to build main distributional properties characterizing thegeneralized $q-$ Quesne quantum algebra, used in physics. Other known resultsin the literature are also recovered as particular cases.

R(p,q)- analogs of discrete distributions: general formalism and application