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Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var

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Answer #1

TOPIC:Properties of expectation,variance and covariance.

Prroblem- 2 Aefine two contnuous pandom vanlab les as 1 we haug E(Rty) + d xdxdy FRIN X X + xp p Ar S+r (x) +E() Provew and s(x+). van (21 2 () +ElY +2xY t X -E () t2 E (R) E E(2 E(). El) tE 2 + E(Ky)E() +2 (x) +van(Y) 2 Cov (x,y) =Van T:var =E (x) -Cov (x) (3J E x y-XE() - E (x). Y tE (x). E() 1 E ()EE E +E (x).EC) E (xY)E (x). E ()E (x).E te ()) (xY)E (x) E CY)indepen dent (4) x and y ane then, Then Cov (x, (xyE (K). E () = E (x). E (Y) - E Cx). E C) E (xy) E (x). Elv) (2ue indupendehny a1,8, 0, Ondtaily Cov (ax tb, ey d) me ac xy t ad x tbe y tb . E (xy) +ad E M) be - EalN-ep -ad ac E 2ac. Cov (x,Y ac (Pr

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