Question

By expanding$\mathit{w}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}$ in a three-variable power series similarto ,$\mathbf{\left(}\mathbf{10.10}\mathbf{\right)}$show that

${\mathit{r}}_{\mathbf{w}}\mathbf{=}\sqrt{{\mathbf{\left(}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{x}}\mathbf{\right)}}^{\mathbf{2}}{\mathbf{r}}_{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{\left(}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{y}}\mathbf{\right)}}^{\mathbf{2}}{\mathbf{r}}_{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{\left(}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{z}}\mathbf{\right)}}^{\mathbf{2}}{\mathbf{r}}_{\mathbf{z}}^{\mathbf{2}}}$

Short answer

Required expressions are,

$\begin{array}{c}\mathbf{E}\left[w\right(x,y,z\left)\right]=w({\mu }_x,{\mu }_y,{\mu }_s)\\ \mathrm{Var}\left(\overline{w}\right)={\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mx}^2\\ {\sigma }_{m\infty }=\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mz}^2}\\ r_w=\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{r}_x^2+{\left(\frac{\partial w}{\partial y}\right)}^2{r}_y^2+{\left(\frac{\partial w}{\partial z}\right)}^2{r}_z^2}\end{array}$

Step-by-step solution

Given Information

The equation,$\left(10.10\right)$ i.e.,role="math" localid="1664361601057" $w(x,y)\cong w({\mu }_x,{\mu }_y)+\left(\frac{\partial w}{\partial x}\right)(x-{\mu }_x)+\left(\frac{\partial w}{\partial y}\right)(y-{\mu }_y)$

Definition of Power series. 

An infinite series is a polynomial with an infinite number of elements that can be thought of as a power series.

Step 3:Expand function w  with Taylor’s series. 

$w(x,y,z)\cong w({\mu }_x,{\mu }_y)+\left(\frac{\partial w}{\partial x}\right)(x-{\mu }_x)+\left(\frac{\partial w}{\partial y}\right)(y-{\mu }_y)+\left(\frac{\partial w}{\partial z}\right)(z-{\mu }_z)$

Calculate mean and variance value of function w.

Calculate expected value.

$\begin{array}{c}\mathbf{E}\left[w\right(x,y,z\left)\right]\cong w({\mu }_x,{\mu }_y,{\mu }_z)+\left(\frac{\partial w}{\partial x}\right)\left[E\left(x\right)-{\mu }_x\right]+\left(\frac{\partial w}{\partial y}\right)\left[E\left(y\right)-{\mu }_y\right]+\left(\frac{\partial w}{\partial z}\right)\left[E\left(z\right)-{\mu }_z\right]\\ =w({\mu }_x,{\mu }_y,{\mu }_s)\end{array}$

Calculate variance.

$\begin{array}{c}\mathrm{Var}\left(\overline{w}\right)=\mathrm{Var}\left[w\right(\overline{x},\overline{y},\overline{z}\left)\right]\\ =\mathrm{Var}[w({\mu }_x,{\mu }_y,{\mu }_z)+\left(\frac{\partial w}{\partial x}\right)(\overline{x}-{\mu }_z)+\left(\frac{\partial w}{\partial y}\right)(\overline{y}-{\mu }_y)+\left(\frac{\partial w}{\partial z}\right)(\overline{z}-{\mu }_z)]\\ ={\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mx}^2\end{array}$

Calculate standard error.

Calculate standard error value.

$\begin{array}{c}{\sigma }_{m\infty }=\sqrt{\mathrm{Var}\left(\overline{w}\right)}\\ =\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mz}^2}\end{array}$

Calculate probable error with confidence interval.

Calculate probable error value.

$\begin{array}{c}{r}_w=I{\sigma }_{mw}\\ =I\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mz}^2}\\ =\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\left(I{\sigma }_{mx}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\left(I{\sigma }_{my}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\left(I{\sigma }_{mz}\right)}^2}\\ =\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{r}_x^2+{\left(\frac{\partial w}{\partial y}\right)}^2{r}_y^2+{\left(\frac{\partial w}{\partial z}\right)}^2{r}_z^2}\end{array}$

Hence, required expressions are,

$\begin{array}{c}\mathbf{E}\left[w\right(x,y,z\left)\right]=w({\mu }_x,{\mu }_y,{\mu }_s)\\ \mathrm{Var}\left(\overline{w}\right)={\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mx}^2\\ {\sigma }_{m\infty }=\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2+{\left(\frac{\partial w}{\partial z}\right)}^2{\sigma }_{mz}^2}\\ r_w=\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{r}_x^2+{\left(\frac{\partial w}{\partial y}\right)}^2{r}_y^2+{\left(\frac{\partial w}{\partial z}\right)}^2{r}_z^2}\end{array}$