2. Let f and F be the pdf and cdf of a continuous random variable X. Suppose that the sample (X1,..., Xn) is not a random sample, but for n 2, the joint pdf of the order statistics ((1), ..., X(n)) can be written as n-1 X (1),... X (n) (x1, xn) = f(xn) I f(xi)/{1 F(xi)}, x1 < x2 < ... < Xn, and 0 otherwise. (a) Suppose that i=1 f(x) = { exp(-x) x > 0, 0 otherwise, and that W = W1, W2,..., Wn are mutually independent. X(1) and Wi = X(i) - X(i-1) for i = 2,...,n. Show that February 22, 2024 (b) Determine the exact distribution of X (n) using the result in (a). (c) Show that X (n) is asymptotically normal. Identify the mean and variance of the normal distribution.