# Question: Let be a pair of independent random variables with the

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).

**View Solution:**## Answer to relevant Questions

Suppose N1is a discrete random variable equally likely to take on any integer in the set {1, 2, 3}. Given that, N1 = n1, the random variable N2 is equally likely to take on any integer in the set {1, 2… n2}. Finally, given ...A random variable, X, has a Gaussian PDF with mean 5 and unit variance. We measure 10 independent samples of the random variable. (a) Determine the expected value of the sample mean. (b) Determine the variance of the ...Suppose Xk is a sequence of zero- mean Gaussian random variables with co-variances described by Cov (Xk, Xm) =ρ |k– m| for some |ρ| < 1. Form the sequence of sample means In this case we are forming the sequence of ...Consider the lottery described. (a) Assuming six million tickets are sold and that each player selects his/ her number independent of all others, find the exact probability of fewer than 3 players winning the lottery. (b) ...Company A manufactures computer applications boards. They are concerned with the mean time before failures (MTBF), which they regularly measure. Denote the sample MTBF as ǔM and the true MTBF as μM. Determine the number of ...Post your question