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Magazine Chapter 4: Q 8E (page 255)
Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{m}}}\)and \({{\bf{Y}}_{\bf{1}}}{\bf{,}}{{\bf{Y}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{Y}}_{\bf{n}}}\)are random variables such that \({\bf{Cov}}\left( {{{\bf{X}}_{\bf{i}}}{\bf{,}}{{\bf{Y}}_{\bf{j}}}} \right)\)exists for\({\bf{i = 1,}}...{\bf{,m}}\)and \({\bf{j = 1,}}...{\bf{,n}}\),and suppose that \({{\bf{a}}_{\bf{1}}}{\bf{,}}{{\bf{a}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{a}}_{\bf{m}}}\)and \({{\bf{b}}_{\bf{1}}}{\bf{,}}{{\bf{b}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{b}}_{\bf{n}}}\) are constants. Show that
\({\bf{Cov}}\left( {\sum\limits_{{\bf{i = 1}}}^{\bf{m}} {{{\bf{a}}_{\bf{i}}}{{\bf{X}}_{\bf{i}}}{\bf{,}}\sum\limits_{{\bf{j = 1}}}^{\bf{n}} {{{\bf{b}}_{\bf{j}}}{{\bf{Y}}_{\bf{j}}}} } } \right){\bf{ = }}\sum\limits_{{\bf{i = 1}}}^{\bf{m}} {\sum\limits_{{\bf{j = 1}}}^{\bf{n}} {{{\bf{a}}_{\bf{i}}}{{\bf{b}}_{\bf{j}}}{\bf{Cov}}\left( {{{\bf{X}}_{\bf{i}}}{\bf{,}}{{\bf{Y}}_{\bf{j}}}} \right)} } \)
Short Answer
If\({X_1},{X_2},...,{X_m}\)and\({Y_1},{Y_2},...,{Y_n}\)are random variables and \(Cov\left( {{X_i},{Y_j}} \right)\)exists for \(i = 1,...,m\) and \(j = 1,...,n\)and \({a_1},{a_2},...,{a_m}\)and \({b_1},{b_2},...,{b_n}\)are constants, then
\(Cov\left( {\sum\limits_{i = 1}^m {{a_i}{X_i},\sum\limits_{j = 1}^n {{b_j},{Y_j}} } } \right) = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{a_i}{b_j}Cov\left( {{X_i},{Y_j}} \right)} } \) .
Step by step solution
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