Explanation

(a)
We have, $Z = aX + bY\$ …(i)
$\Rightarrow$ $\overline{Z} = a\overline{X} + b\overline{Y}$ …(ii)
$Z - \overline{Z} = a\left( X - \overline{X} \right) + b(Y - \overline{Y})$
$\Rightarrow$ $\left( Z - \overline{Z} \right)^{2} = a^{2}\left( X - \overline{X} \right)^{2} + b^{2}\left( Y - \overline{Y} \right)^{2} + 2ab\left( X - \overline{X} \right)(Y - \overline{Y})$
$\Rightarrow$ $\frac{1}{n}\sum_{}^{}\left( Z - \overline{Z} \right)^{2} = a^{2}\frac{1}{n}\ \sum_{}^{}\left( X - \overline{X} \right)^{2} + b^{2}\frac{1}{n}\ \sum_{}^{}\left( Y - \overline{Y} \right)^{2} + 2ab\frac{1}{n}\sum_{}^{}{\left( X - \overline{X} \right)(Y - \overline{Y})}$
$\Rightarrow$ $\sigma_{Z}^{2} = a^{2}\sigma_{X}^{2} + b^{2}\sigma_{Y}^{2} + 2ab\ cov\ (X,\ Y)$
$\Rightarrow$ $\sigma_{Z}^{2} = a^{2}\sigma_{X}^{2} + b^{2}\sigma_{Y}^{2} + 2ab\ r\sigma_{X}\sigma_{Y}$
$\left\lbrack \therefore\frac{cov(X,Y)}{\begin{array}{r}
\sigma_{X}\sigma_{Y} \\
\
\end{array}} = r \right\rbrack$