29. Let X(i) denote ith smallest of a sample of size n from a continuous distribution function F . Also, let U(i) denote the ith smallest from a sample of size n from a uniform (0, 1) distribution.

(a) Argue that the density function of U(i) is given by

[Hint: In order for the ith smallest of n uniform (0, 1) random variables to equal t , how many must be less than t and how many must be greater? Also, in how many ways can a set of n elements be broken into three subsets of respective sizes i − 1, 1, and n − i?]

(b) Use part

(a) to show that E[U(i)] = i/(n + 1). [Hint: To evaluate the resulting integral, use the fact that the density in part

(a) must integrate to 1.]

(c) Use part

(b) and Problem 28a to conclude that E[F (X(i))]=i/(n + 1).

Transcribed Image Text:

fu) (t) = n! (n-i)!(i-1)! - (1)", 0