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r te joint probabilthy denokty Aunchan: de 2 2 For šome pesihve, real constant ,finol h condlitiànal expectedl value of X givPLEASE HELPPPPP thank you!!

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Answer #1

f(x, y) f (y)

f(y) = | f(x,y)dr

= rac{2y}{k^2}

f(x|Y =y) = rac{rac{2}{k^2}}{rac{2y}{k^2}} = rac{1}{y}

E(X|Y=y) =int _0^yrac{x}{y}dx=rac{y}{2}

E(X) =int _0^kint _0^yrac{2x}{k^2}dxdy=rac{k}{3}

E(X^2) = int _0^kint _0^yrac{2x^2}{k^2}dxdy=rac{k^2}{6}

E(Y)

k2 12

k2 k2ry drdyA 12

r = Cov(X,Y)/(sd(X) * sd(Y))

Cov(X,Y) = E(XY) - E(X)E(Y)

= k^2/4 - k/3 * 2k/3 = k^2/36

sd(X) = sqrt(E(X^2) - (E(X))^2) = sqrt( k^2/6 - k^2/9) = k/sqrt(18)

sd(Y) = sqrt(k^2/2 - 4k^2/9) = k/sqrt(18)

r = rac{rac{k^2}{36}}{ rac{k}{sqrt{18}} * rac{k}{sqrt{18}} } = rac{1}{2}

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