Order Statistics. Let X1, X2, , Xn be a random sample of size n from X, a random variable having distribution function F(x). Rank the elements in order of increasing numerical magnitude, resulting in X(1), X(2), , X(n), where X(1) is the smallest sample element (X(1) = min{X1, X2, , Xn}) and X(n) is the largest sample element (X(n) = max{X1, X2, , Xn}). X(i) is called the ith order statistic. Often the distribution of some of
the order statistics is of interest, particularly the minimum and maximum sample values. X(1) and X(n), respectively. Prove that the cumulative distribution functions of these two order statistics, denoted respectively by FX(1) (t) and FX(a)(t)are FX(1)(t) = [1 €“ 1 F(t)]n Fx(a)(t) = [F(t)]n Prove that if X is continuous with probability density function f (x), the probability distributions of X(1) and X(n) are

Sx (0) = n[1 - F()}"-/) fx,(1) = n[F{1)]*!S)