(a)
For a fair coin, p = 1/2
Var(X) = Var(Y) = Var(Z) = p(1-p) = (1/2)(1 - 1/2) = 1/4
Cov(W, V) = Cov(2X - 3Y + Z, X - 2Y - Z)
= Cov(2X, X - 2Y - Z) + Cov(- 3Y, X - 2Y - Z) + Cov(Z, X - 2Y - Z)
= Cov(2X, X) + Cov(2X, - 2Y) + Cov(2X, - Z)
+ Cov(- 3Y, X) + Cov(- 3Y, - 2Y) + Cov(- 3Y, - Z)
+ Cov(Z, X) + Cov(Z, - 2Y) + Cov(Z, - Z)
= 2Cov(X, X) - 2*2 Cov(X, Y) + 2 * (-1) Cov(X, Z)
+ (-3)*Cov(Y, X) + (-3) * (-2)Cov(Y, Y) + (-3)*(-1)Cov(Y, Z)
+ Cov(Z, X) + (-2)Cov(Z, Y) + (-1)Cov(Z, Z)
= 2 Var(X) + 6 Var(Y) - Var(Z)
= 2 * (1/4) + 6 * (1/4) - (1/4)
= 7/4
(b)
Var(W) = Var(2X - 3Y + Z) = 22Var(X) + (-3)2Var(Y) + Var(Z)
= 4 * (1/4) + 9 * (1/4) + (1/4) = 14/4 = 7/2
Var(V) = Var(X - 2Y - Z) = Var(X) + (-2)2Var(Y) + (-1)2Var(Z)
= (1/4) + 4 * (1/4) + (1/4) = 6/4 = 3/2
Corr(W, V) = Cov(W, V) /
= (7/4) /
= 0.7638
Extra: Let X, Y, Z be results of three independent tosses of a fair die. (a)...
I need help on 6.26 and 6.28 please!
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