An economic consultant for a fast-food chain has been given a random sample of the chains service workers. In her model, A, the length of time y between the time the worker is hired and the time a worker quits has an exponential distribution with parameter 1, p(y | ) = exp(y). For the purposes of this problem, assume that no one is ever laid off or fired. The only way of leaving employment is by quitting. You can also assume that no one ever has more than one spell of employment with the companyif they quit, they never come back. In this problem we consider the case in which the consultants data consist entirely of complete spells; that is, for each individual t in the sample the consultant observes the length of time, yt , between hiring and quitting.

(a) Express the joint density of the observables and find a sufficient statistic vector for .

(b) Show that the conjugate prior distribution of has the form s2 2(), and provide an artificial data interpretation of (s2, ).

(c) Using the prior density in (b), express the kernel of the posterior density. Show that the posterior distribution of has a gamma distribution of the form s2 2(). Express s2 and in terms of the sufficient statistics from (a) and ( 1, 2) from (b).