A) \[{{a}^{2}}\sigma _{X}^{2}+{{b}^{2}}\sigma _{Y}^{2}+2abr\,{{\sigma }_{X}}{{\sigma }_{Y}}\]
B) \[{{a}^{2}}\sigma _{X}^{2}+{{b}^{2}}\sigma _{Y}^{2}-2abr\,{{\sigma }_{X}}{{\sigma }_{Y}}\]
C) \[2abr\,{{\sigma }_{X}}{{\sigma }_{Y}}\]
D) None of the above
Correct Answer: A
Solution :
We have, \[Z=aX+bY\] ...(i) \[\Rightarrow \] \[\overline{Z}=a\overline{X}+b\overline{Y}\] ...(ii) From Eqs. (i) and (ii), \[Z-\overline{Z}=a(X-\overline{X})+b(Y-\overline{Y})\] \[\Rightarrow \]\[{{(Z-\overline{Z})}^{2}}={{a}^{2}}{{(X-\overline{X})}^{2}}+{{b}^{2}}(Y-\overline{Y})\] \[+2ab(X-\overline{X})(Y-\overline{Y})\] \[\Rightarrow \] \[\frac{1}{n}\Sigma {{(Z-\overline{Z})}^{2}}={{a}^{2}}\frac{1}{n}\Sigma {{(X-\overline{X})}^{2}}\] \[+{{b}^{2}}\frac{1}{n}\Sigma {{(Y-\overline{Y})}^{2}}\] \[\Rightarrow \] \[\sigma _{z}^{2}={{a}^{2}}\sigma _{X}^{2}+{{b}^{2}}\sigma _{Y}^{2}+2ab\operatorname{cov}(X,Y)\] \[\Rightarrow \] \[\sigma _{z}^{2}={{a}^{2}}\sigma _{X}^{2}+{{b}^{2}}\sigma _{Y}^{2}+2ab\,\,r\,{{\sigma }_{X}}\,{{\sigma }_{Y}}\] \[\left[ \because \frac{\operatorname{cov}(X,Y)}{{{\sigma }_{X}}{{\sigma }_{Y}}}=r \right]\]
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