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 $$ g(X) \Rightarrow g(\mu)+g^{\prime}(\mu) \cdot(X-\mu) $$ 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 \(l\) is random, then resistance will also be a random variable related to \(I\) by \(R=v / I\). If \(\mu_{l}=20\) and \(\sigma_{l}=.5\), calculate approximations to \(\mu_{R}\) and \(\sigma_{R}\).

Step by step solution

01

Analyze Taylor Series Approximation

The given Taylor series approximation: \( g(X) \Rightarrow g(\mu)+g^{\prime}(\mu) \cdot(X-\mu) \) implies that \( g(X) \) is approximately linear around \( \mu \). This approximation allows us to apply the properties of linear functions to estimate the expected value and variance of \( g(X) \).

02

Use Properties of Expectation for Linear Functions

For a linear function \( aX+b \), the expected value \( E(aX+b) \) is given by \( aE(X) + b \). Substituting \( a = g'(\mu) \) and \( b = g(\mu) \), we get \( E(g(X)) \approx g(\mu) + g'(\mu)E(X - \mu) \). Since \( E(X - \mu) = 0 \), \( E(g(X)) \approx g(\mu) \).

03

Use Properties of Variance for Linear Functions

The variance of a linear function \( aX+b \) is \( a^2 \text{Var}(X) \). So, for \( g(X) \approx g(\mu) + g'(\mu)(X - \mu) \), the variance is \( \text{Var}(g(X)) \approx (g'(\mu))^2 \text{Var}(X) \). Therefore, \( V(g(X)) \approx (g'(\mu))^2 \sigma^2 \).

04

Calculate Approximation for \( \mu_R \)

Given \( R = v/I \), we approximate \( g(I) = v/I \) around \( \mu_l \). Using \( g(I) = v/I \), the derivative is \( g'(I) = -v/I^2 \). Hence, \( g(\mu_l) = v/\mu_l \), and \( g'(\mu_l) = -v/\mu_l^2 \). Thus, \( E(R) \approx v/\mu_l \approx g(\mu_l) \). If \( v \) is fixed, this simplifies to \( \mu_R = v/\mu_l = v/20 \).

05

Calculate Approximation for \( \sigma_R \)

The variance approximation for \( R = v/I \) is \( V(R) \approx (g'(\mu_l))^2 \sigma_l^2 \). Substituting the values, \( g'(\mu_l) = -v/\mu_l^2 \), and \( \sigma_R \approx |g'(\mu_l)| \sigma_l = \frac{v}{\mu_l^2} \times 0.5 = \frac{v}{400} \times 0.5 \). Thus, if \( v \) is known, use this to find \( \sigma_R \).

06

Numerical Calculations Using Given \( \mu_l \) and \( \sigma_l \)

Inserting specific values, if \( v = 100 \), then \( \mu_R = \frac{100}{20} = 5 \) and \( \sigma_R = \frac{100}{400} \times 0.5 = 0.125 \). This shows the needed computations for specific cases of \( v \).

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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, often symbolized as \(E(X)\), is a fundamental concept in probability and statistics. It represents the long-term average or mean value of a random variable when the experiment is repeated many times. To compute the expected value for linear functions, we use the formula for linear transformation: \(E(aX + b) = aE(X) + b\).
Applying this formula to our function \(g(X)\), which is approximated as \(g(\mu) + g'(\mu) \cdot (X - \mu)\), we see that the expected value becomes \(E(g(X)) \approx g(\mu)\) because \(E(X - \mu) = 0\).
This simplification highlights that if the random variable fluctuates around its mean within small deviations, the expected value of our Taylor-approximated function stays close to the value at the mean.

Variance

Variance measures how much a set of values, like our random variable \(X\), is spread out over a range. Mathematically, it's denoted by \(\text{Var}(X)\). Variance is crucial for understanding the variability or inconsistency in a dataset or a random variable.
When dealing with a linear approximation through a Taylor series, the variance of a transformation \(g(X) \approx g(\mu) + g'(\mu) \cdot (X - \mu)\) is calculated using: \(\text{Var}(aX + b) = a^2 \text{Var}(X)\).
For our function \(g(X)\), the variance becomes \(\text{Var}(g(X)) \approx (g'(\mu))^2 \sigma^2\).
Here, \(\sigma^2\) is the variance of the random variable \(X\), and \(g'(\mu)\) is the derivative at the mean. This formula shows how a small change in \(X\) corresponds to a proportional adjustment in the variance of \(g(X)\), scaled by the square of the derivative.

Random Variables

Random variables are variables that can take on different values, each with an associated probability. They're used to model uncertain phenomena where outcomes cannot be predicted with certainty.
In our context, the random variable \(X\) is central to understanding how the function \(g(X)\) behaves as \(X\) changes. Typically, random variables are characterized by their distribution, which might be concentrated around the mean or more spread out.

• **Discrete Random Variables** - Take specific (countable) values.
• **Continuous Random Variables** - Take a continuous range of values, like all numbers in an interval.

The behavior and properties of \(g(X)\) depend significantly on how \(X\)'s values are distributed. If \(X\) has values confined to a region where \(g(X)\) is almost linear, the approximations we derive are more accurate.

Linear Functions

Linear functions are essential mathematical tools used to describe relationships between variables with a constant rate of change. These functions have the form \(y = aX + b\), where \(a\) and \(b\) are constants.
In the context of Taylor series approximation, linear functions become particularly useful. The approximation \(g(X) \approx g(\mu) + g'(\mu) \cdot (X - \mu)\) transforms the original function into a simple linear form when \(X\) is near \(\mu\). This makes computation straightforward because both expected value and variance for linear functions are easier to handle compared to non-linear expressions.
The beauty of this approach is its simplicity: by treating \(g(X)\) as linear around \(\mu\), we can leverage basic properties of linear functions like additivity for expected value \(E\) and scale-invariance for variance \(Var\). This yields quick estimations for otherwise complex functions when the random variable fluctuates near the average.