Application of a new discrete distribution

In epidemiology, an infection lasting n weeks may be monitored by taking weekly serum samples. If tests on samples are independent Bernoulli trials with probability q of correctly testing positive, the apparent duration of infection ( from the first positive test to the last positive test inclusive) may be less than n weeks. This distribution of apparent length also arises when plants in a row of n each have a probability q of germinating, for example. This distribution is shown to be related to that of the number of tails obtained when tossing a coin until two heads are obtained, in a maximum of n tosses. The properties of the 'apparent length' distribution are described, and some compounded (mixed) distributions that can be derived from it are also discussed. The distribution was used to estimate the underlying distribution of the duration of infection, in a longitudinal study of infections of children. The methodology was also used to estimate the proportion of infectious episodes that were not detected. It can be similarly used to correct episode durations and rates in longitudinal studies in which episodes of any kind are detected by regular sampling.