Suppose that X and Y, the proportions of an 8-hour workday that two quality control inspectors actually spend on performing their assigned duties, have joint probability density

a. Find the marginal probability distributions, f₁(x) and f₂(y).

b. Verify that

c. Find the conditional probability distributions, f₁(x Ι y) and f₂(y Ι x).

d. Verify that

e. Are X and Y correlated? Are X and Y independent?

f. The proportion d of “dead” time (i.e., time when no assigned duties are performed) for the two attendants is given by the relation D = 1 - (X + Y)/2. Find E(D) and V(D). Within what limits would you expect D to fall?

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f(x, y) = (x + y if 0 sIs 1;0 < ys 1 to elsewhere