Every time drivers take to the road, and with each mile that they drive, exposes themselves and others to the risk of an accident. Insurance premiums are only weakly linked to mileage, however, and have lump-sum characteristics largely. The result is too much driving, and too many accidents. In this paper, we introduce some useful theoretical results for Pay-As-You-Drive in Automobile insurances. We consider a counting process and also find the distribution of discounted collective risk model when the counting process is non-homogeneous Poisson.
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