Let be the number of losses random variable and let X be the severity random variable. If there is a deductible of d imposed, there are two ways to modify X. One is to create , the amount paid per loss:

In this case, the appropriate frequency distribution continues to be .

An alternative approach is to create , the amount paid per payment:

In this case, the frequency random variable must be altered to reflect the number of payments. Let this variable be . Assume that, for each loss, the probability is that a payment will result. Further assume that the incidence of making a payment is independent of the number of losses. Then, , where is 0 with probability and is 1 with probability v. Probability generating functions yield the relationships in Table C.1.

	Parameters for
Poisson
ZM Poisson
Binomial
ZM binomial
Negative binomial
ZM negative binomial
ZM logarithmic

The geometric distribution is not presented as it is a special case of the negative binomial with . For zero-truncated distributions, the same formulas are still used as the distribution for will now be zero modified. For compound distributions, modify only the secondary distribution. For ETNB secondary distributions, the parameter for the primary distribution is multiplied by as obtained in Table C.1, while the secondary distribution remains zero truncated (however, .

There are occasions on which frequency data are collected that provide a model for . There would have to have been a deductible d in place and therefore v is available. It is possible to recover the distribution for , although there is no guarantee that reversing the process will produce a legitimate probability distribution. The solutions are the same as in Table C.1, only now .

Now suppose that the current frequency model is , which is appropriate for a deductible of d. Also suppose that the deductible is to be changed to . The new frequency for payments is and is of the same type. Then use Table C.1 with .