| Abstrakt: |
The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter a is a positive integer.The result is a generalization of Lundberg (1940).Consider the distribution ofwhere X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution functionand Nis an integer valued random variable with probability functionThen the distribution function of Sis given byIf MX(t), MN(t)and MS(t)are the associated moment generating functions, then |
