n Let X1, ..., X be i.i.d. continuously distributed real random variables with Lebesgue-density denoted dPx1 by f = and cumulative distribution function F. We define d = X(1) = min{X : 1 i n}, X (2) = min ({X; : 1 i n} \ {X(1)}) = the second smallest of X, 1, Xn, X(n) = max{X : 1 i n}. Then X(1), ..., X(n) are also random variables which fulfill X(1) are called order statistics. (a) Show that the joint density of the order statistics is given by X (n). Collectively, these [n! II" f(xi) x1 < x2 < ... < Xn, X(1),...,X(n) (x1, ..., Xn) 0, otherwise. (b) Show that the r-th element X (r) (with 1 r n) of the order statistics has the density - x()(x) = n(, 1) f(x) F(x)'(1 F(x))"". fx