In an automobile plant, two tasks are performed by robots. The first entails welding two joints; the second, tightening three bolts. Let X denote the number of defective welds and Y the number of improperly tightened bolts produced per car. Past data indicate the joint probability mass function as shown in the table. y p( x, y) 1 2 3 0 0.840 0.030 0.020 0.010 X 0.060 0.010 0.008 0.002 2 0.010 0.005 0.004 0.001 a) Is this a valid probability mass function? Justify your answer. b) Compute the marginal pmf of X and Y. c) What is the probability that there are defective welds? d) What is the expected number of improperly tightened bolts produced per car? e) What is the probability that there will be 1 improperly tightened bolt given there is 1 defective weld? f) What is the probability that there will be 1 defective weld given there is 1 improperly tightened bolt