The law of a sum of independent identically distributed claims with an independent Poisson distribution count of parameter . If the claim moment-generating function is , the aggregate transform is wherever finite. It has an atom at zero when the claims are positive. This is the fixed-time law of a Compound Poisson process.
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The Compound Poisson distribution is a statistical distribution that arises in the context of counting events that occur randomly over time or space, where each event results in a random, typically discrete, amount of "impact" or "size." It combines two probabilistic processes: 1. **Poisson Distribution**: This component models the number of events that occur within a fixed interval (time or space) under the assumption that these events happen independently and at a constant average rate.