The management of a bank must decide whether to install a commercial loan decision-support system (an online management information system) to aid its analysts in making commercial loan decisions. Past experience shows that X, the additional number (per year) of correct loan decisions—accepting good loan applications and rejecting those that would eventually be defaulted—attributable to the decision-support system, and Y, the lifetime (in years) of the decision-support system, have the joint probability distribution shown in the table at the top of the page.
a. Find the marginal probability distributions, p₁(x) and p₂(y).
b. Find the conditional probability distribution, p₁(x Ι y).
c. Given that the decision-support system is in its third year of operation, find the probability that at least 40 additional correct loan decisions will be made.
d. Find the expected lifetime of the decision-support system, i.e., find E(Y).
e. Are X and Y correlated? Are X and Y independent?
f. Each correct loan decision contributes approximately $25,000 to the bank’s profit. Compute the mean and standard deviation of the additional profit attributable to the decision-support system.

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X=x 10 20 30 40 50 60 70 80 90 1 .001 .002 .002 .025 .040 .025 .005 .005 .005 .005 .010 .075 .100 .075 .050 .030 .030 .025 Y=y 3 .025 .050 .080 .050 .080 .040 .030 4 .001 .002 .005 .010 .025 .010 .003 .001 .001 .002 .005 .005 .020 .030 .015 2.