Let X take the value 0, 1, or 2 depending on whether there are no customers, between 1 and 10 customers, and more than 10 customers in the regular (manned)

checkout lines of a supermarket. Let Y be the corresponding variable for the self checkout lines. An extensive study undertaken by the management team of the supermarket resulted in the following conditional distributions of Y given X = x, and the marginal distribution of X:

(a) Use the Law of Total Probability to find the marginal PMF of Y.

(b) Use the multiplication rule for joint probabilities to tabulate the joint distribution of X and Y.

y 0 1 2 PYX-0(V) 0.85 0.10 0.05 X 0 1 2 PYX=1(Y) 0.30 0.45 0.25 Px(x) 0.20 0.50 0.30 PYIX=2(y) 0.20 0.35 0.45