2.20 Let X(r) be the rth-order statistic of a random sample of size n from a cdf FX.

(a) Verify that P(X(r)  t) ¼

Pnk

¼r n

k

 

[FX(t)]k[1  FX(t)]nk.

(b) Verify the probability density function of X(r) given in (2.6.4) by differentiation of the result in (a).

(c) By considering P(X(r)>t=n) in the form of (a), find the asymptotic distribution of X(r) for r fixed and n!1if FX(x) is the uniform (0, 1)

distribution.