91Ó°ÊÓ

A function's derivative is defined as the function's instantaneous rate of change at a given location.

The following is true:

\(\begin{array}{l}\frac{{\partial h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial {\mu _1}}} = \frac{\partial }{{\partial {\mu _1}}}\left( {{\mu _4} \cdot \left( {\frac{1}{{{\mu _1}}} + \frac{1}{{{\mu _2}}} + \frac{1}{{{\mu _3}}}} \right)} \right)\\ = - \frac{{{\mu _4}}}{{\mu _1^2}}\\\frac{{\partial h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial {\mu _2}}} = \frac{\partial }{{\partial {\mu _2}}}\left( {{\mu _4} \cdot \left( {\frac{1}{{{\mu _1}}} + \frac{1}{{{\mu _2}}} + \frac{1}{{{\mu _3}}}} \right)} \right)\\ = - \frac{{{\mu _4}}}{{\mu _2^2}}\\\frac{{\partial h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial {\mu _3}}} = \frac{\partial }{{\partial {\mu _3}}}\left( {{\mu _4} \cdot \left( {\frac{1}{{{\mu _1}}} + \frac{1}{{{\mu _2}}} + \frac{1}{{{\mu _3}}}} \right)} \right)\\ = - \frac{{{\mu _4}}}{{\mu _3^2}},\\\frac{{\partial h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial {\mu _4}}} = \frac{\partial }{{\partial {\mu _4}}}\left( {{\mu _4} \cdot \left( {\frac{1}{{{\mu _1}}} + \frac{1}{{{\mu _2}}} + \frac{1}{{{\mu _3}}}} \right)} \right)\\ = \frac{1}{{{\mu _1}}} + \frac{1}{{{\mu _2}}} + \frac{1}{{{\mu _3}}}\end{array}\)

Second-order partial derivatives are also

\(\begin{array}{l}\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _1^2}} = \frac{{2{x_4}}}{{x_1^3}}\\\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _2^2}} = \frac{{2{x_4}}}{{x_2^3}}\\\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _3^2}} = \frac{{2{x_4}}}{{x_3^3}}\\\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _4^2}} = 0\end{array}\)

The following is correct based on the values provided in the preceding exercise.

\(\begin{array}{l}E(Y) \approx h\left( {{\mu _1}, \ldots ,{\mu _n}} \right) + \frac{1}{2}\sigma _1^2\left( {\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _1^2}}} \right) + \frac{1}{2}\sigma _2^2\left( {\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _2^2}}} \right)\\ + \frac{1}{2}\sigma _3^2\left( {\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _3^2}}} \right) + \frac{1}{2}\sigma _4^2\left( {\frac{{{\partial ^2}h\left( {{\mu _1},{\mu _2},{\mu _3},{\mu _4}} \right)}}{{\partial \mu _4^2}}} \right)\\ = 26 + 0.12 + 0.0356 + 0.0338\\ = 26.1894\end{array}\)

Therefore, the value of \(E(Y) \approx 26.1894\).