Abstract
In this paper we introduce some significant counting distributionsin risk theory. The first one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models
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