Bolts manufactured for a certain purpose may be classified as acceptable (suitable for the intended purpose), downgraded (unsuitable for the intended purpose but acceptable for a different purpose), or scrap (unsuitable for any purpose). In a lot of 500 bolts, let X be the number that are downgraded and let Y be the number that are scrap. Assume that the joint probability mass function of X and Y is given in the following table.

a. Find the marginal probability mass function of X.
b. Find the marginal probability mass function of Y.
c. Are X and Y independent? Explain.
d. Find μX and μY.
e. Find σX and σY.
f. Find Cov(X, Y).
g. Find ρ(X, Y).

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3 0 0.06 0.03 0.01 0.00 0.00 1 0.06 0.08 0.04 0.02 0.00 2 0.04 0.05 0.12 0.06 0.03 3 0.00 0.03 0.07 0.09 0.06 4 0.00 0.00 0.02 0.06 0.07