# Question

Let be a pair of independent random variables with the same exponential PDF,

fXi (x) = exp(– x) u( x) i = 1, 2

Define Y1, Y2 to be the order statistics associated with the Xi. That is, Y1 = min (X1, X2) and Y2 = min (X1, X2).

(a) Find the marginal PDFs of Y1 and Y2, fY1 (y1) and fY2 (y2).

(b) Find the joint PDFs of Y1 and Y2, fY1, y2 (y1, y2).

(c) Find the MAP estimator of Y2 given Y1 = y1

(d) Find the ML estimator of Y2 given Y1 = y1

(e) Find the LMMSE estimator of Y2 given Y1 = y1

(f) Find the MSE of each estimator in (c), (d), and (e).

fXi (x) = exp(– x) u( x) i = 1, 2

Define Y1, Y2 to be the order statistics associated with the Xi. That is, Y1 = min (X1, X2) and Y2 = min (X1, X2).

(a) Find the marginal PDFs of Y1 and Y2, fY1 (y1) and fY2 (y2).

(b) Find the joint PDFs of Y1 and Y2, fY1, y2 (y1, y2).

(c) Find the MAP estimator of Y2 given Y1 = y1

(d) Find the ML estimator of Y2 given Y1 = y1

(e) Find the LMMSE estimator of Y2 given Y1 = y1

(f) Find the MSE of each estimator in (c), (d), and (e).

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