Question

Let X,Y be uniformly distributed in the rectangle defined by −3 < x−y < 3, 1 < x + y < 5. Find the marginal density fX(x) and E(Y|X).In the same situation find Cov(X,Y ).

(3) Let X, Y be uniformly distributed in the rectangle defined by -3 < x-y<3, Find the marginal density fx(x) and E(Y|X). In the same situation find Cov(X, Y). 1<x+y<5.

Answer 1

194)X_y=-3 X+YES 2-=3 (-1,2) +91 Now, the area of the rectangle is, . - (1-45+(4-15 (2-4²+(-1-12 = N18 V8 - 12 .

So, the joint PDF of (x,y) is,.. fx, y (x,y) = area of the rectangle . -3< 2-y<3 1<x+y<5 1. o, ow Now, for -Kxkol fx @ = I do = (x+3+(1–] 2 +3 X + 3 . 1- 2

For ka<2 dy 5-x- 1+x - انا falle) = stdine = 5-*=*** = t = Ś 12 1-X For 22n2u 5-7 fa(x) = s = 5*5**3 = 27 2 4 2-3 Now, -1<2<1 Bydlen) = ( f (y ) = 1 Jylo 1-X<y< 7+3 l= fx; y(1,4) fx las e ols who <x<2 INDLY 5-2 2<xcu I 2<244 -3 <y< 5-X <n <, 1- 7 +3 24 +1). DICK<2 ) 1-<<5- 2<x24 204-X) x-3<425-8

པls) - ] /g01༧) ༧༔ E (7] ༩） 5- 6- ༡ ་་ ་ +3 2(ཀ +| ་བ |- - (༡༠-༠• Ie (ཁག - * ] x+། ༥， ༡ 25-| 16 ༤ ༡ 46x +2+ལྟ(25-1= ༥༠༣ ཀ ཁ༡） E ༥(x 41） ་ (25 - 1༠༧ – ༡ +s») |༠- t - ༡ + 2– 2 + = ཉ - ༡

NNNN EQ) = E(E (7[x)) = E(9-x) = 9 - E(X) 4 - اقا *{**1* $(9) + 4- )

Now) E(Y) = 9-# Now, E(XY) = b[E (x y1x)]: = E [X E(Y1x3] = E[x.(9'- x)] = E@x-x) = 9 E(X) – E6%).

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