Consider three ping-pong balls numbered 1, 2, and 3. Two balls are randomly selected with replacement. If the sum of the two resulting numbers exceeds 4, two balls are again selected. This process continues until the sum is at most 4. Let X and Y denote the last two numbers selected. Possible (X, Y) pairs are {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)}.

a. Determine p_X,Y(x,y).
b. Determine p_Y|X(y|x).
c. Determine E(Y|X = x). Is this a linear function of x?
d. Determine E(X|Y = y). What special property of p(x, y) allows us to get this from (c)?
e. Determine V(Y|X = x).