Suppose that X1,... ,Xn are i.i.d. random variables with a continuous distribution function F, symmetric about 0. Let R + ni = be the rank of |Xi | among |X1|,... , |Xn|, and Si = sign(Xi), for i =...

Suppose that X1,... ,Xn are i.i.d. random variables with a continuous distribution function F, symmetric about 0. Let R + ni = be the rank of |Xi | among |X1|,... , |Xn|, and Si = sign(Xi), for i = 1,... ,n. Show that the two vectors R + n = (R + n1 ,... ,R+ nn) ⊤ and Sn = (S1,... ,Sn) ⊤ are stochastically independent. R + n can assume all possible n! permutations of 1,... ,n), with the common probability 1 n! and Sn takes on all possible 2 n sign-inversions (i.e., (±1,... , ±1)⊤) with the common probability 2−n. What happens if F is not symmetric about 0?