For three categorical variables A, B, and C:

a. When C is jointly independent of A and B, show that A and C are conditionally independent, given B.

b. Prove that mutual independence of A, B, and C implies that A and B are (i)

marginally independent and (ii) conditionally independent, given C.

c. Suppose that A is independent of B and that B is independent of C. Does this imply that A is independent of C? Explain.