The second random variable is the amount of the loss, given that a loss has occurred. For example, the hospital charges for an overnight hospital stay would be much lower than the charges for an extended hospitalization. The amount of loss is often referred to as the severity and the probability distribution for the amount of loss is called the severity distribution. By combining the frequency distribution with the severity distribution we can determine the overall loss distribution.
Example: Consider a car owner who has an 80% chance of no accidents in a year, a 20% chance of being in a single accident in a year, and no chance of being in more than one accident in a year. For simplicity, assume that there is a 50% probability that after the accident the car will need repairs costing 500, a 40% probability that the repairs will cost 5000, and a 10% probability that the car will need to be replaced, which will cost 15,000. Combining the frequency and severity distributions forms the following distribution of the random variable X, loss due to accident:
