29. Instead of conditioning the confidence sets 8 E S( X) on a set C, consider a randomized procedure which assigns to each point x a probability IJ!(x) and makes the confidence statement 8 E S( x) with probability IJ!( x) when x is observed.*

(i) The randomized procedure can be represented by a nonrandomized conditioning set for the observations (X, U), where U is uniformly distributed on (0,1) and independent of X, by letting C = {(x, u) : u < Ii'(x)}. (ii) Extend the definition of relevant and semirelevant subsets to randomized conditioning (without the use of U). (iii) Let 8 E S(X) be equivalent to the statement X E A(8). Show that Ii' is positively biased semirelevant if and only if the random variables Ii'(X) and I A(8)( X ) are positively correlated, where I A denotes the indicatorof the set A.