In the present paper, certain random damage models are examined, such as the generalized MARKOV-POLY A (GMP), the Quasi-Binomial, and the Quasi-Hypergeo-metric, in which an integer random variable N is reduced to B. Following JANAEDAN (1973 b) who has characterized the Multivariate Hypergeometric distribution in terms of the Multinomial, we have shown that under the GMP damage model, the distributions of N and B both belong to the family of the generalised POLYA-EGGENBERGER (GPE) distributions. We have also shown that the damage model can be uniquely identified as the GMPD given that B and N belong to the same GPE family. A physical interpretation of the result is given