Question: 6. Let X1, X2, . . . , Xn be independent random variables, each having distribution function F and density function f . Find the

6. Let X1, X2, . . . , Xn be independent random variables, each having distribution function F and density function f . Find the distribution function of U and the density functions of U and V , where U = min{X1, X2, . . . , Xn} and V = max{X1, X2, . . . , Xn}. Show that the joint density function of U and V is fU,V (u, v) = n(n − 1) f (u) f (v)

F(v) − F(u)

n−2 if u < v.

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