I was hoping someone can show me the proof for
V(y1-Y2) = V(y1)+V(y2)- 2 Cov(Y1,Y2)
or
V(y1+Y2) = V(y1)+V(y2)+ 2 Cov(Y1,Y2)
I understand the proof for V(y1+Y2) = V(y1) + V(Y2) for independent but im having trouble inferring for the dependent.
I was hoping someone can show me the proof for
V(y1-Y2) = V(y1)+V(y2)- 2 Cov(Y1,Y2)
or
V(y1+Y2) = V(y1)+V(y2)+ 2 Cov(Y1,Y2)
I understand the proof for V(y1+Y2) = V(y1) + V(Y2) for independent but im having trouble inferring for the dependent.
$\displaystyle V(y_1+y_2)=E( [(y_1+y_2)-\overline{(y_1-y_2)}]^2)$
but
$\displaystyle
\overline{(y_1-y_2)}=\overline{y_1} + \overline{y_2}
$
so:
$\displaystyle V(y_1+y_2)=E( [ (y_1-\overline{y_1}) +(y_2-\overline{y_2})]^2)$ $\displaystyle =E((y_1-\overline{y_1}) ^2) + 2E((y_1-\overline{y_1}) (y_2-\overline{y_2}) )+E((y_2-\overline{y_2}) ^2)$
CB