91Ó°ÊÓ

Consider an rv \(X\) with mean \(\mu\) and standard deviation \(\sigma\), and let \(g(X)\) be a specified function of \(X\). The first-order Taylor series approximation to \(g(X)\) in the neighborhood of \(\mu\) is $$ g(X) \approx g(\mu)+g^{\prime}(\mu) \cdot(X-\mu) $$ The right-hand side of this equation is a linear function of \(X\). If the distribution of \(X\) is concentrated in an interval over which \(g(\cdot)\) is approximately linear [e.g., \(\sqrt{x}\) is approximately linear in \((1,2)]\), then the equation yields approximations to \(E(g(X))\) and \(V(g(X))\). a. Give expressions for these approximations. [Hint: Use rules of expected value and variance for a linear function \(a X+b .]\) b. If the voltage \(v\) across a medium is fixed but current \(I\) is random, then resistance will also be a random variable related to \(I\) by \(R=v / I\). If \(\mu_{I}=20\) and \(\sigma_{I}=.5\), calculate approximations to \(\mu_{R}\) and \(\sigma_{R}\).

Step by step solution

01

Determine Taylor Series Linearization

The Taylor series linearization of the function \( g(X) \) around \( \mu \) is given by: \[g(X) \approx g(\mu) + g^{\prime}(\mu) \cdot (X - \mu)\] This is a linear approximation.

02

Calculate Expected Value Approximation

The rule for expected value of a linear function \( aX + b \) is:\[E(aX + b) = aE(X) + b\]Applying this to the linear approximation:\[E(g(X)) \approx E(g(\mu) + g^{\prime}(\mu) \cdot (X - \mu)) = g(\mu) + g^{\prime}(\mu) \cdot (E(X) - \mu)\]Since \( E(X) = \mu \), this simplifies to:\[E(g(X)) \approx g(\mu)\]

03

Calculate Variance Approximation

For variance, the rule for a linear function \( aX + b \) is:\[V(aX + b) = a^2 V(X)\]Applying this to our linear approximation:\[V(g(X)) \approx V(g^{\prime}(\mu) \cdot (X - \mu)) = (g^{\prime}(\mu))^2 V(X)\]Since \( V(X) = \sigma^2 \), we have:\[V(g(X)) \approx (g^{\prime}(\mu))^2 \sigma^2\]

04

Define the Random Variable for Resistance

Given \( R = v/I \) and the current \( I \) is a random variable with \( \mu_I = 20 \) and \( \sigma_I = 0.5 \). Let's express \( R \) in terms of \( I \):\[g(I) = \frac{v}{I}\]Here, the function \( g(I) = \frac{v}{I} \) is to be linearized around \( \mu_I = 20 \).

05

Derive the First Derivative

Compute the first derivative of \( g(I) \):\[g^{\prime}(I) = \frac{-v}{I^2}\]Evaluating at \( I = \mu_I = 20 \):\[g^{\prime}(20) = \frac{-v}{20^2} = \frac{-v}{400}\]

06

Calculate the Resistance Mean Approximation

The approximation for the expected value of \( R \) is:\[E(R) \approx g(\mu_I) = \frac{v}{20}\]since the constant term in the linear approximation around the mean provides the approximation for the mean.

07

Calculate the Resistance Variance Approximation

Using the variance approximation:\[V(R) \approx \left(\frac{-v}{400}\right)^2 \cdot (0.5^2) = \frac{v^2}{160000} \cdot 0.25 = \frac{v^2}{640000}\]Thus, the standard deviation is:\[\sigma_R = \frac{v}{800}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value

The expected value is a fundamental concept in probability and statistics, representing the average or mean value of a random variable across numerous trials. For a random variable \( X \), the expected value, denoted as \( E(X) \), indicates the long-term average outcome if the experiment that produces \( X \) were to be repeated many times.When dealing with an application of the Taylor series, as in the exercise, one can use the properties of expected value for linear functions. If you consider a function expressed as \( aX + b \), the expected value can be calculated using:

• \( E(aX + b) = aE(X) + b \)

Applying this to the Taylor series approximation of \( g(X) \), we have:

• \( E(g(X)) \approx g(\mu) + g^{\prime}(\mu) \times (E(X) - \mu) \)

Given that \( E(X) = \mu \), this simplifies to \( E(g(X)) \approx g(\mu) \). Thus, the expected value of the function approximates to its value at the mean of \( X \). This simplification is especially useful when \( g(X) \) is approximately linear around \( \mu \).

Variance

Variance is another critical statistical measure that tells us how spread out the values of a random variable are from its expected value. In simpler terms, it measures the variability or dispersion of the data points.For a linear function of the form \( aX + b \), the variance can be determined using the formula:

• \( V(aX + b) = a^2 V(X) \)

In the context of a Taylor series approximation of \( g(X) \), this becomes:

• \( V(g(X)) \approx (g^{\prime}(\mu))^2 V(X) \)

This indicates that the variance of the function \( g(X) \) is proportional to the variance of \( X \), scaled by the square of the derivative of \( g \) at the average \( \mu \). The variance tells us how much the approximated function value can deviate assuming \( X \) has a narrow distribution around \( \mu \).

Linearization

Linearization is a technique used to approximate a complex function by a linear function, making calculations simpler while remaining close to the original function's behavior in a narrow interval. In situations described by the Taylor series, linearization becomes particularly useful.For a function \( g(X) \), a first-order linearization around the mean \( \mu \) of \( X \) can be formulated as:

• \( g(X) \approx g(\mu) + g^{\prime}(\mu) \, (X - \mu) \)

Here, \( g^{\prime}(\mu) \) is the derivative of \( g \) evaluated at \( \mu \). This approximate linearly tracks small changes about \( \mu \), allowing us to understand the behavior of \( g(X) \) near this average value. When a function is approximately linear over the interval of interest, this can facilitate calculations of the expected value and variance, which ideally remain simple and easy to apply in various scenarios, such as approximating resistances and voltages as in your exercise.

Random Variables

Random variables are foundational to statistical analysis. These are variables whose outcomes depend on random phenomena. For example, a random variable can represent the result of a dice roll, which can vary each time you roll.In our exercise, the current \( I \) is treated as a random variable. The randomness implies that its value cannot be predicted exactly, but it has an expected value (mean) \( \mu_I = 20 \) and standard deviation \( \sigma_I = 0.5 \). Since resistance \( R = v/I \) relies on a random \( I \), \( R \) also becomes a random variable. Hence, resistances derived under varying current conditions must be analyzed as random variables, using tools like expected value and variance to comprehend their dispersion and predict long-term behaviors effectively. Understanding the characteristics of random variables underpins the reliable approximation of those expressions.