Question

X and Y are JGRVs with my = -1, my = 0, o3 = 4, oi = , 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?

Answer 1

Ques: хүч 1) BUND (-1,0,4, Vyte) yoient palf of X ky Sny mily) ky-> 20 U-lan on 9² f 2 the values peeting fmp (mign 2.2.1 ек 62 63 Folyo yan et pl'atoyleyi (4-4p? 2p( 265 ) ) <PC. L. (1-x2) (+2) -2*** () ()}] () y) 20-%) x 2x J314 empl 3 frente +(ay) KER, YER then 2) Covarian bln. Xory = lou (x4) co ox by 2 x P x J 2 %) -Yo for the conditional; / Nux, 0) My + See (x-llo) density uk whee + 2 2 11 = 0 +(33) Yz (x+1) *(X+3) TA og? (1-5) (í-4) = 366 4) 7. [ + (x - 3) MMSE = var lyt Lo. Bias (ů )= EC4) the [Biasly?j?

x=n yo find undles med ġ = 9 MMS & ustimate y given (m) Y given X = n is : Ely/x-n ? Jy f/4) dy this function is of form gry) & axt 6 let MSEL hlaib) say [4-984273 8[ -fax + b) Then en by =-14 the function hea,b) is minimized it a= Cou(x14 = 1/2 Voulx ) b = 6(74) – €(X) = 0 -2.41)== a = -Yy. bar/4 hlaib)= a xt 6 = -1 4X -14 -14(UH) 801 X = n = y So, mmsef of y giun X = n the ever of linear MMSE is et q-y-ax-b (x+1) YtY4 Thus, the linear MM se extimator of Yginent is ŷ t = Coulx,y) VW (x ) = 0+ -Y [x+1] [x+1] YL which we chave derived 2 27 14 chave aber MSE already € [4 - 9]= 11-p") Van (4) lineau MM SE is उ (1-74) /4 3/16

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