91影视

The standard deviation is a measurement of a collection of values' variance or dispersion. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation suggests that the values are dispersed over a larger range.

The three random variables \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}\) are independent of the random variable \({{\rm{X}}_{\rm{4}}}\)since they are linked in parallel across a battery with voltage \({{\rm{X}}_{\rm{4}}}\). The exercise's outcome,

\({\rm{E(Y)\gg h}}\left( {{{\rm{\mu }}_{\rm{1}}}{\rm{,}}{{\rm{\mu }}_{\rm{2}}}{\rm{,}}{{\rm{\mu }}_{\rm{3}}}{\rm{,}}{{\rm{\mu }}_{\rm{4}}}} \right)\)

Hence, the following is true

\(\begin{array}{c}{\rm{E(Y)\gg E}}\left( {{{\rm{X}}_{\rm{4}}}\left( {\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{1}}}}}{\rm{ + }}\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{2}}}}}{\rm{ + }}\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{3}}}}}} \right)} \right)\\{\rm{ = E}}\left( {{{\rm{X}}_{\rm{4}}}\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{1}}}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{4}}}\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{2}}}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{4}}}\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{3}}}}}} \right)\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{E}}\left( {{{\rm{X}}_{\rm{4}}}} \right){\rm{ \times E}}\left( {\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{1}}}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{4}}}} \right){\rm{ \times E}}\left( {\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{2}}}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{4}}}} \right){\rm{ \times E}}\left( {\frac{{\rm{1}}}{{{{\rm{X}}_{\rm{3}}}}}} \right)\\{\rm{ = 120 \times }}\frac{{\rm{1}}}{{{\rm{10}}}}{\rm{ + 120 \times }}\frac{{\rm{1}}}{{{\rm{15}}}}{\rm{ + 120 \times }}\frac{{\rm{1}}}{{{\rm{20}}}}\\{\rm{ = 26}}\end{array}\)

(1): because of independence.

Therefore, the expected value is \({\rm{E(Y) = 26}}{\rm{.}}\)

The partial derivatives are required to determine estimated variance. The following is true:

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

The approximate variance can be computed as follows

\(\begin{array}{c}V(Y) \approx {\left( {\frac{{\partial h}}{{\partial {\mu _1}}}} \right)^2} \cdot \sigma _1^2 + {\left( {\frac{{\partial h}}{{\partial {\mu _2}}}} \right)^2} \cdot \sigma _2^2 + {\left( {\frac{{\partial h}}{{\partial {\mu _3}}}} \right)^2} \cdot \sigma _3^2 + {\left( {\frac{{\partial h}}{{\partial {\mu _4}}}} \right)^2} \cdot \sigma _4^2\\\mathop = \limits^{(2)} {( - 1.2)^2} \cdot 1 + {( - 0.5333)^2} \cdot 1 + {( - 0.3)^2} \cdot 1.5 + {(0.2167)^2} \cdot 4\\ = 2.6783\end{array}\)

(2): substitute given values.

Therefore, the approximate standard deviation of given random variable is \({{\rm{\sigma }}_{\rm{Y}}}{\rm{\gg }}\sqrt {{\rm{2}}{\rm{.6783}}} {\rm{ = 1}}{\rm{.64}}{\rm{.}}\)