5. Let X take the value 0 if a child between 4 and 5 years of age uses no seat belt, 1 if he or she uses a seat belt, and 2 if it uses a child seat for short-distance car commutes.

Also, let Y take the value 0 if a child survived a motor vehicle accident and 1 if he or she did not. Accident records from a certain state suggest the following conditional PMFs of Y given X = x and marginal distribution of X:1

(a) Use the table of conditional PMFs of Y given X = x to conclude whether or not X and Y are independent.
Justify your answer.

(b) Make a table for the joint PMF of (X,Y), showing also the marginal PMFs, and use it to conclude whether or not X and Y are independent. Justify your answer.

y 0 1 X 0 1 2 PYX-0) 0.69 0.31 PYX-1(y) 0.85 0.15 PY X-2(y) 0.84 0.84 0.16 Px(x) 0.54 0.17 0.29