Question

Problem 2 X and Y are JGRVs with mx = -1, my = 0, ož = 4, 0} = 1, and p = -1 1. What is the joint pdf of X and Y? 2. What is the covariance matrix of X and Y? 3. What is f (y|x)? 4. What is the MMSE estimate of Y when X is given? What is the resulting error? Problem 3 In problem 2 define Z = X + Y - 3 and W = 2X – 2Y + 1. 1. What is the covariance matrix for Z and W? 2. What is the joint pdf of Z and W? 3. What is the MMSE estimate of Z when W is given? What is the resulting error?

Answer 1

problem a Giver my=-1, my = 0 4 y ² - ''A 4 joint pold (x. y) is 2. ny ny Z IST-yr ((x + 1) (y - o) + (x+1) 1 xe ETA YA 13 YA W%$********y+3) €43(x2+1 +23 +1649 +4wy+4y) TS3 4 covariance matrix is tary G2 А + (1/1) is normal parametey! E (YX = W) = my + Sx y (x-mu) = 0 - 77 (4+1) 2 --5640

War Cytte ) = (1 pr.) 2 -16 4 ty/u) - YER 4 64 1 MASE - E (x ) = x + 1 Resulting error = 7 van C&J = problem Given that 2=xty-3 2x-2y + var(Z) = van (x+y-3) = Vas (x+y) W= var (u) of varly) + 2 Couluy) 4 + 4 + 2xPony 13 4 var (2) S var (W) = 4 varu) + a varly) - 2 COV(2x, 2y) = 16+ 1-2x 4 Cor(ay) = 18 E (Z) = E + tly) - 3 =-1+0-3 = -4 8 (W) =-2+ = -1 CON (2, W) = 8-1/2 = 18/2 Z. w are also bivariate normal 126 ( 4 (2+4)² (Wit)? 4 (2+4) (wt w (2,W)= 317 18 26x13x 18 e 2 + 13 ZW

A X 26 MMSE of zu E (2/2) = E(2) + 2 varice) (w - E(W)) vaw) - ris 18 (wti) --4+ Tera (w+) =-4+ 5 (W+1) Resulting erro Svar (E (2/6) 25 var(w) = 25X18 12X12 1) ولی کی 12 SH 4