The claim size for a certain insurance policy is a Pareto random variable with density function , . The parameter is taken to be an observation from the random variable whose prior distribution has the density function , . A claim of is observed. X ? ? ? ? 2 3 2 f x x ? ? ? ? ? 0 x ? ? ? ? ? 2 1 ? ? ? ?1? ?? ?3 X ?

a) Find the density function for the posterior distribution

b) Find the cumulative distribution function for the posterior distribution.

c) Calculate the posterior probability that . (You can use the results in (b)). 2? ?

d) Find the expected value of the posterior distribution.