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(a) Consider the joint probability density fX,Z (x, z) = e-z for 0 ≤ x ≤ z and fX,Z (x, z) = 0 otherwise. Find the pair x, z of values that maximize this density. Find the marginal density fZ (z) and find the value of z that maximizes this.

(b) Let fX,Z,Y (x, z, y) be y2e-yz for 0 ≤ x ≤ z, 1 ≤ y ≤ 2 and be 0 otherwise. Conditional on an observation Y = y, find the joint MAP estimate of X, Z. Find fZ|Y (z|y), the marginal density of Z conditional on Y = y, and find the MAP estimate of Z conditional on Y = y.

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