Question

11 a12 13 11 021 C11 C12 C13 a31 a32 a33 1)13 123 C31 C32 C33

Answer 1

$4.~E(W)=(1,-2,1)E(Y)=(1,-2,1)\begin{pmatrix} \mu\\ -\mu\\ \mu \end{pmatrix}=\mu+2\mu+\mu=4\mu\\ Disp(W)=(1,-2,1)\begin{pmatrix} 1\\ -2\\ 1 \end{pmatrix}\sigma^2+\phi^2~since~Y~and~\epsilon ~are~independent.\\ =6\sigma^2+\phi^2\\ 5.~Cov(W,Y)=\left(Cov(W,Y_1),Cov(W,Y_2),Cov(W,Y_3) \right )\\ =(Var(Y_1),-2Var(Y_2),Var(Y_3))=(\sigma^2,-2\sigma^2,\sigma^2),~since~Y_i's~are~independent\\$

$6.~Corr(W,Y)=\left(Cov(W,Y_1)/\sqrt{Var(W)Var(Y_1)},Cov(W,Y_2)/\sqrt{Var(W)Var(Y_1)},Cov(W,Y_3)/\sqrt{Var(W)Var(Y_1)} \right )\\ =(\sigma^2/\sqrt{6\sigma^2+\phi^2},-2\sigma^2/\sqrt{6\sigma^2+\phi^2},\sigma^2/\sqrt{6\sigma^2+\phi^2}),\\ ~since~Var(Y_1)=Var(Y_2)=Var(Y_3)=1\\ 7.~Cov(V)=Cov\begin{pmatrix} WY_1\\ WY_2\\ WY_3 \end{pmatrix}=\begin{pmatrix} Var(WY_1) &Cov(WY_1,WY_2) & Cov(WY_1,WY_3) \\ Cov(WY_1,WY_2) & Var(WY_2) & Cov(WY_2,WY_3) \\ Cov(WY_1,WY_3) &Cov(WY_2,WY_3) & Var(WY_3) \end{pmatrix},\\ ~where,~Y=\begin{pmatrix} Y_1\\ Y_2\\ Y_3 \end{pmatrix}\\ =\begin{pmatrix} Var(Y_1^2-2Y_1Y_2+Y_1Y_3) &Cov(Y_1^2-2Y_1Y_2+Y_1Y_3,Y_1Y_2-2Y_2^2+Y_2Y_3) &Cov(Y_1^2-2Y_1Y_2+Y_1Y_3, Y_1Y_3-2Y_2Y_3+Y_3^2)\\ &Var(Y_1Y_2-2Y_2^2+Y_2Y_3) & Cov(Y_1Y_2-2Y_2^2+Y_2Y_3,Y_1Y_3-2Y_2Y_3+Y_3^2)\\ & & Var(Y_1Y_3-2Y_2Y_3+Y_3^2) \end{pmatrix}$