Let XI, X2 , X3 be independent N(I', 0' 2 ) variates.

(a) Show that for IVI :S: 11'/6 we may write X( 3) = X( 2) + S cos V +v'3Ssin V, X( 1 ) = X( 2) - S cos V + v'3Ssin V, where

(b) Show that

so that V or Y are in 1: 1 correspondence with the sample coefficient of skewness.

(c) By transfonning from (XC I ) , X( 2) , X( 3 ) to (S, X( 2) , V), show that V is unifonn over (-11'/6, 11'/6).