Choose a prior distribution that captures your opinion about the parameters of this problem. Use a simulation method to calculate the first two moments of the joint posterior distribution on the parameters of this model. Use a simulation method to obtain plots of the approximate marginal posterior densities of the two parameters. An experiment on the reliability of pumps at nuclear power plants recorded failures of 10 pumps monitored over various periods of time. Pump failures for pumps at the Farley 1 nuclear plant.
Times are in thousands of hours.
One possible model for these data is a hierarchical Poisson model. Failures are assumed to occur according to independent Poisson processes with each pump having its own failure rate . These rates are modeled as independent draws from a gamma distribution with density proportional to . Suppose it is reasonable to give a gamma prior distribution with and and to assume that . Use a simulation method to compute posterior means and standard deviations for the and for .