In a study with n = 7814 participants, the frequency of syncopes was investigated. As a result, y = 822 syncopes were observed during that study. The mean observation time in the study was 17 years. The total person-time at risk was T = 133164 years.

1. Identify an appropriate distribution family to model the situation above and derive the maximum likelihood estimate for the incidence rate based on that distribution family. Provide a detailed explanation on your modelling and calculations for full credit.
2. Let the estimate in 1 be given by = y / T . Give a 95% confidence interval CI for this estimate.
3. Calculate the maximum likelihood estimate for the log-incidence rate = log and construct the corresponding 95% confidence interval CI for . Further, use an appropriate transformation of CI in order to derive the 95% confidence interval for . Compare your results with the results obtained in 2.