4. Let X1,X2, . . . ,Xn be iid RVs with common Pareto DF f (x) = ασα/xα+1, x > σ

where α>0, σ>0. Show that

(a) X(1) and (X(2)/X(1), . . . ,X(n)/X(1)) are independent,

(b) X(1) has Pareto (σ,nα) distribution, and

(c)

n j=1 n(X(j)/X(1)) has PDF f (x) =

xn−2e−αx

(n−2)!

, x > 0.