Resumo: Apresentamos uma generalização da distribuição binomial associada com números positivos. Esta generalização representa sistemas com correlação entre os eventos. Nosso enfoque é baseado em funções geradoras e satisfaz os vínculos requeridos para uma interpretação probabilista da distribuição deformada. Mostramos vários exemplos analíticos, discutimos a conexão da nossa proposta com a deformação da distribuição binomial de Laplace-de Finetti, discutimos a regra de Leibniz para os vários exemplos e apresentamos modelos simples, tipo urna, para os vários casos. Estes exemplos também nos permitem explorar, analiticamente, o comportamento destas distribuições quando o número de eventos n é muito grande. Calculamos analiticamente em diversos casos a distribuição limite para n tendendo a infinito. Estudamos também para estes sistemas com correlação entre os eventos a extensividade de diversas formas entrópicas (para n grande) e encontramos resultados analíticos inesperados em alguns casos.
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