1)     Renewal processes: describe the distribution of the number of events in a given time interval and/or the distribution of inter-event times.

2)     Mortality models: illustrate the usefulness of Markov chains in the modelling of mortality. Describe the Binomial model of the mortality of a group of identical individuals (subject to no other decrements between two given ages).

3)     Survival, sickness, marriage and other simple models: described via/as applications of Markov processes. Derive Kolmogorov forward and backward equations for a Markov process with time independent and time/age dependent transition and solve them in simple cases.

4)     Likelihood functions: describe the Cox model for proportional hazards and use the partial likelihood est imate (with standard errors) to estimate its parameters. Reduce the Binomial likelihood function to a function of a single parameter, under the assumption of uniform distribution of deaths or constant force of mortality, and estimate this single parameter.

5)     2-state models: derive the likelihood function for a 2-states model with states Alive and Dead, under Cox, Binomial and Poisson models.

6)     Multi-state models: describe an appropriate Markov multi-state model for a system with multiple transfers, derive the likelihood function in a Markov multi-state model with data and use the likelihood function to estimate the parameters (with standard errors).

7)     Comparison: describe the advantages and disadvantages of the multi-state model versus the Binomial model, including consistency, efficien cy, simplicity of the estimators and their distributions.

8)     Survival functions: estimate a survival function using the Kaplan-Meier or Nelson-Aalen method.

9)     Statistical tests: explain statistical tests of crude estimates for comparing with a standard table:  chi-square test, standardised deviations test, cumulative deviation test, sign test, grouping of signs test and the serial correlation test to testing the adherence of graduation data.

10)    Exposed to risk: Connection between estimation of transition intensities and exposed to risk (central and initial exposed to risk), exact calculation of the central exposed to risk, census approximations to central exposed to risk (trapezium approximation).

11)    Graduation: describe the process of gradu ation and test for the smoothness of graduated estimates.