2. (12 points) A form of the Pareto distribution has density function f(x) = fax-(a+1) x ≥ 1 otherwise 0 where a > 0 is

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2 12 Points A Form Of The Pareto Distribution Has Density Function F X Fax A 1 X 1 Otherwise 0 Where A 0 Is 1 (157.25 KiB) Viewed 26 times

2. (12 points) A form of the Pareto distribution has density function f(x) = fax-(a+1) x ≥ 1 otherwise 0 where a > 0 is a parameter. We observe n iid realizations of the Pareto random variable. (a) Show that a Gamma distribution with shape parameter a and rate parameter b (or scale parameter 1/b) is a conjugate prior for this Pareto distribution. (b) Suppose we use an exponential prior with rate parameter A. Find the posterior mean in terms of A and the data.