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Problem 4 (10 pts). (a) For two independent variables x and y, show that E(xy) = E(x) E(y) and E(ax + by) = aE(x) + bE(y), where a and b are constants. (b) Pos- sibly using your result from (a), complete the derivation of Var(x) = E ((x -M)?) = 02 on slide 18-20, Lecture 6.Standard deviation of the mean Show var (x) = E (x - 1)2) =62 (Prob. 2, p775, Boas). Recall x = -E x;. Similarly, we may write u = = Eil. :. X - 1 = => ( x1 -14) : (x - u ) 2 = n 2 [ (X1 -H) (2) = 1 2 ) ( X 1 - 1 ) ( x ; - 1 ) (3) ij 1 = ( X1 - H) ( x, - 14) + ( 27 -14) (2; -14) 14) = [ (1 -4 )2 + [ ( 9 -14) (x, -14) (5 ) : E((x - 1) 2 ) = E -2 X ( x1 -1)2 + [ ( x1-1)(x, -14) (6) THE [,(G - H)2 + [ ( x 1 - H) (x, - 4) (7 )Standard deviation of the mean Show var (x) = E (x -1)2) = 62 (Prob. 2, p775, Boas). . . . : E((x - M) 2) = - ( X1 - 14 ) 2 + ) ( 29 -14) (2 , -14) (1) = E [ (X 1 - 14 ) 2 | + 2 E [ ( x 1 - 1 ) ( x , - H ) 121 = n 2 LE[ (x1 - 14) (x, - 14)] (3) 1 = n2 no2 + 15 n : Ziti E[(x; - M)] E[(x, - M) ] (4) 02 1 S + S 0 .0 n n2 Ziti (5) = (6 ) n . E (( X - 1) 2 ) = (7)