1. The Pareto distribution is a continuous power law distribution widely used to describe many datasets in social sciences and economics. If Z ~ Pareto(uo, a) then P(Z > z) = where uo is the minimum value for Z, assumed known, and o is the Pareto inder. (a) Find the density of Z and then argue that Z does not belong to the exponential family. (b) While Z does not belong to the exponential family, Y = log Z does. What are the canonical parameter and the cumulant function for Y? (c) Using the cumulant, find the mean / and the variance of Y. What is the variance function V(u)? Now obtain the canonical link for the distribution of Y. (d) Show that Y - log po ~ Exponential(a). How would you regress Z on a set of predictors X to estimate the Pareto index using glm