Question: 10.5.6 A random variable has a Pareto distributionwith parameters a , b (a > 0, b > 0) if its density is a ,x >

10.5.6 A random variable has a Pareto distributionwith parameters a , b (a > 0, b >

0) if its density is a

,x > 0. f (x;a1

b) = b ( l + x/b)a+l Let XI,. . . X, be a random sample from the Pareto distributionwith density f(z;1 ,l).
(i) Find the limiting distribution of random variable U, = n X l Z n . [Hint: Find cdf F(x)o f X i , and then determine cdf of U, in terms of F(x).](i i) Show that V, = Xn:, does not have a proper limiting distribution; specifically, limn-m P{ V, 5 t } = 0 for every t . (iii) Find the limiting distribution of V,/n.

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