2. (a) The number of claims follows a negative binomial distribution with parameters [3 and r, where [3 is unknown and r is known. You wish to estimate B based on n observa- tions, where x is the mean of these observations. Determine the maximum likelihood estimate of B . [7 marks] (b) You are given the following information, about a credibility model: First Observation Unconditional Probability Bayesian Estimate of Second Observation Determine the credibility estimate of the second observation, given that the first ob- servation is 1. [9 marks] (0) The number of defects in a magnetic tape has a Poisson distribution with unknown mean 9. The prior distribution of 9 is a gamma distribution with on = 3, B = 1. Five rolls of magnetic tape are tested for defects and it is found that the number of defects is 2,2,6,0,3. If we use the squared error loss function, what is the Bayes estimate of 9 '? [9 marks] Total: 25 marks