AbstractIn this study we introduce the Polya-Aeppli-Lindley distribution as amixed Polya- Aeppli distribution with Lindley mixing distribution. Someproperties, probability mass function and recursion formulas are given.Then we define a stochastic counting process with Polya-Aeppli-Lindleydistribution. An equivalent definition as a pure birth process and someadditional properties are given. As an application, we introduce a Polya-Aeppli- Lindley risk model, and analyze some properties, ruin probabilityand deficit at the time of ruin.
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