C. R. Rao [Information and the accuracy attainable in the estimation of statistical parmeters. Bull. Calcutta Math. Soc. 37, 81-91 (1945)] pointed out that the statistical distribution which is specified by a set of n real parameters \(\Theta =(\theta^ 1,\theta^ 2,...,\theta^ n)\) constructs an n-dimensional Riemannian space under the suitable regularity conditions. Recently, an \(\alpha\)-metric concept has been proposed which is able to generalize the conception of statistical distance. However, discrete distributions have not been researched by using the \(\alpha\)-metric concept. We propose a method for an application of the \(\alpha\)-metric concept to the binomial distribution and the trinomial one, which are discrete distributions, and study the construction of this parameter space.