Chapter 15: Q7P (page 775)

Equation $(10 .12)$ isonly an approximation (but usually satisfactory). Show, however, that if you keep the second order termsin, $(10.10)$ then
role="math" localid="1664364127028" $\bar{w} = w (\bar{x}, \bar{y}) + \frac{1}{2} (\frac{\partial^{2} w}{\partial x^{2}}) σ_{x}^{2} + \frac{1}{2} (\frac{\partial^{2} w}{\partial y^{2}}) σ_{y}^{2}$ .

Short Answer

Expert verified

Required expression is: $\bar{w} = w (\bar{x}, \bar{y}) + \frac{1}{2} (\frac{\partial^{2} w}{\partial x^{2}}) σ_{x}^{2} + \frac{1}{2} (\frac{\partial^{2} w}{\partial y^{2}}) σ_{y}^{2}$

Step by step solution

Given Information

The equation, $(10.10)$ i.e., $w (x, y) ≅ w (μ_{x}, μ_{y}) + (\frac{\partial w}{\partial x}) (x - μ_{x}) + (\frac{\partial w}{\partial y}) (y - μ_{y})$

The equation $(10.12)$ , i.e., $\bar{w} = w (\bar{x}, \bar{y})$

Definition of Expectation

The expected value is a generalisation of the weighted average, commonly known as expectation, in probability theory.

Expand using Taylor’s series.

Write the expression using Taylor’s Series.

$\begin{matrix} w (x, y) ≅ w (μ_{x}, μ_{y}) + (\frac{\partial w}{\partial x}) (x - μ_{x}) + (\frac{\partial w}{\partial y}) (y - μ_{y}) \\ + \frac{1}{2} [\frac{\partial^{2} w}{\partial x^{2}} {(x - μ_{x})}^{2} + 2 \frac{\partial^{2} w}{\partial x \partial y} (x - μ_{x}) (y - μ_{y}) + \frac{\partial^{2} w}{\partial y^{2}} {(x - μ_{y})}^{2}] \\ (w (x, y)) ≅ w (μ_{x}, μ_{y}) + (\frac{\partial w}{\partial x}) (E (x) - μ_{x}) + (\frac{\partial w}{\partial y}) (E (y) - μ_{y}) \\ + \frac{1}{2} [\frac{\partial^{2} w}{\partial x^{2}} E {(x - μ_{x})}^{2} + 2 \frac{\partial^{2} w}{\partial x \partial y} E [(x - μ_{x}) (y - μ_{y})] + \frac{\partial^{2} w}{\partial y^{2}} E {(x - μ_{y})}^{2}] \end{matrix}$

Calculate expected value.

Solve for expected value.

$\begin{matrix} (E (x) - μ_{x}) = (E (y) - μ_{y}) \\ = 0 \\ E {(x - μ_{x})}^{2} = σ_{x}^{2} \\ E {(x - μ_{y})}^{2} = σ_{y}^{2} \end{matrix}$

Simplify further.

$\begin{matrix} E [(x - μ_{x}) (y - μ_{y})] = 0 \\ \bar{w} = w (\bar{x}, \bar{y}) + \frac{1}{2} (\frac{\partial^{2} w}{\partial x^{2}}) σ_{x}^{2} + \frac{1}{2} (\frac{\partial^{2} w}{\partial y^{2}}) σ_{y}^{2} \end{matrix}$

Hence, required expression is:. $\bar{w} = w (\bar{x}, \bar{y}) + \frac{1}{2} (\frac{\partial^{2} w}{\partial x^{2}}) σ_{x}^{2} + \frac{1}{2} (\frac{\partial^{2} w}{\partial y^{2}}) σ_{y}^{2}$

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Short Answer

Step by step solution

Given Information

Definition of Expectation

Expand using Taylor’s series.

Calculate expected value.

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Equation (10.12)isonly an approximation (but usually satisfactory). Show, however, that if you keep the second order termsin,(10.10)thenrole="math" localid="1664364127028" w¯=w(x¯,y¯)+12(∂2w∂x2)σx2+12(∂2w∂y2)σy2.

Short Answer

Step by step solution

Given Information

Definition of Expectation

Expand using Taylor’s series.

Calculate expected value.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Electric Charge Field and Potential

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Fundamentals of Physics

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Study anywhere. Anytime. Across all devices.