5. Let −1/9 < c < 1/9 be a constant. Let p(x, y), the joint probability mass function of the random variables X and Y , be given by the following table: y x −1 0 1 −1 1/9 1/9 − c 1/9 + c 0 1/9 + c 1/9 1/9 − c 1 1/9 − c 1/9 + c 1/9

(a) Show that the probability mass function of X+Y is the convolution function of the probability mass functions of X and Y for all

c.

(b) Show that X and Y are independent if and only if c = 0.