Summary: Substantial work has been dedicated in recent years to the construction of families of continuous distributions obtained by applying a modulation factor to a base symmetric density so as to obtain non-symmetric variant forms, often denoted skew-symmetric distributions. All this development has dealt with the case of continuous variables, while here we extend the formulation to the discrete case; moreover, some of the statements are of general validity. The results are illustrated with an application to the distribution of the score difference in sport matches.
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