Problem 3: Bounds on entropy * * Consider a vector X" of n random variables X, with / e [1; n] whose pdf is px (note that the random variables are not necessarily i.i.d.). [Q1] Prove that for any i E [1; n] H(X") = H({X/);#i) + H(Xi\\{X;);#). where {Xiliz - {Xiljenin \\ {X/} = (X1, ... . Xi-1, Xit1. ..., Xn). [Q2] For n 2 2, prove that n H (X1 . . . ., Xn) 2 CH (X/{X;lixi). i=1 [Q3] Prove that NIK (H (X1X2) + H(X2X3) + H(X1X3)) 2 H(X1X2X3). (Hint: sum the identities H (X1X2X3) = H({X/}i#) + H(Xi|{X;);#) proved in the first part and use the second part.)