ECE 153 Spring 2018 Problem 1: Let X and Y be two independent random variables according to the following densities. fX(x) = { e−x if x ≥ 0, 0 otherwise, fY (y) = { 1 if 0 ≤ y ≤ 1, 0 otherwise. (1) (a) Find the pdf of U = max{X,Y }. (b) Find the conditional pdf of X given the event {X ≥ Y }. (c) Find the joint cdf of X and U , FU,X(u, x). 1 2 Problem 2: Let X and Y be two random variables with joint pdf f(x, y) = { xy c , 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ x+ y ≤ 1 0 otherwise (2) (a) Find the value of c. (b) Find the conditional pdf fX|Y (x|y). (c) Find the MMSE estimator of X given Y . (d) Find the CDF of Z = E(X|Y ) in terms of CDF of Y . (e) Find the MMSE linear estimator of X given Y . 3 4 Problem 3: Let Xn = αZn−1 + βZn+1 for n ≥ 1, where Z0, Z1, Z2, . . . are i.i.d. ∼ N (0, 1). Let Yn = Xn +Wn, where Wn are i.i.d. N (0, 1), independent of Xn. (a) Is the process {Xn} Gaussian? Justify your answer. (b) Find the mean and autocorrelation functions of Xn. (c) Find autocorrelation functions of Yn and cross correlation function RXY . (d) Is Xn wide-sense stationary? Justify your answer. (e) Is Xn strict-sense stationary? Justify your answer. (f) Find MMSE estimator of Xn given Xn−2. (g) Find MMSE estimator of Xn given Yn. 5 6 7 8 Some Useful Facts: • ∫ 10 y(1− y)4dy = 130 • ∫ 10 y3(1− y)2dy = ∫ 10 y2(1− y)3dy = 160 • ∫ 10 y2(1− y)2dy = 130 • ∫ x0 te−tdt = 1− (x+ 1)e−x • The best linear MMSE estimate is Xˆ = Cov(X,Y ) σ2Y (Y − E(Y )) + E(X). • Let X(t), t ∈ R, be a WSS process input to a LTIS with impulse response h(t): X(t) Y (t)h(t) If the system is stable, then the input X(t) and output Y (t) are jointly WSS with: 1. E(Y (t)) = H(0)E(X(t)) 2. RY X(τ) = h(τ) ∗RX(τ) 3. RY (τ) = h(τ) ∗RX(τ) ∗ h(−τ) RX(τ) RY (τ)h(τ) h(−τ) RY X(τ) 9
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