In general business insurance, the maximum accumulated amount of losses retained by the insured under deductible policy modifications is usually set as part of the terms and conditions of the policy documents. However, the risk retention of the insured under deductible is limited for every loss event and on a yearly arrangement. A policy without deductible coverage modifications may encounter excessive large claim amount. The insurer only takes over payment of claims given that the aggregate limit is exceeded. This paper develops an analytical framework for evaluating the effect of structural properties of dirac-delta on insurance risk variables with deductible clauses. The aim is to derive models for the moments and variance of insurance claim size in a loss event. In order to achieve this and create analytically sound and useful theoretical basis of investigating actuarial risk functions, the general properties of dirac-delta is first examined in respect of probability density function after some operations. We then obtained insurance claim severity and variance models for an arbitrary policy in general insurance business under deductible coverage modifications which is meant to disapprove nuisance claims and control problem of moral hazard. The results have been clearly stated and proved as part of our findings which may have significant implications on policy underwriting efficiency and decisions.