Problem 1 (CONDITIONING ENTROPY AND MUTUAL INFORMATION) (a) Let the three discrete random variables X, Y, Z be related by Z = X 7 Y, Where X, Y E {0, . . . ,m 1}, and the subtraction is modulo m. 1. Compare H(X|Y) and H(Z). 2. When is H(X|Y) equal tO H(Z)? 3. Assume the equality condition you found in part (b) is indeed satised, and Z has uniform distribution over {0, . . . ,m 1}. What can you say about I(X; Y)? (b) If X, Y, and Z are joint random variables, prove the following inequalities and nd conditions for equality. 1. H(X, Y, Z) H(X,Y) g H(X, Z) H(X). 2. I(X;Z|Y) 2 I(Z;Y|X) 7112,10 +I(X;Z). (c) If X, Y, and Z are joint random variables, give examples of X, Y, and Z such that 1. I(X;Y|Z) I(X;Y)