91Ó°ÊÓ

Let \(X_{1}, \ldots, X_{n}\) be independent rv's with mean values \(\mu_{1}, \ldots\), \(\mu_{n}\) and variances \(\sigma_{1}^{2}, \ldots, \sigma_{n}^{2}\). Consider a function \(h\left(x_{1}, \ldots\right.\), \(\left.x_{n}\right)\), and use it to define a new rv \(Y=h\left(X_{1}, \ldots, X_{n}\right)\). Under rather general conditions on the \(h\) function, if the \(\sigma_{i} \mathrm{~s}\) are all small relative to the corresponding \(\mu_{i} \mathrm{~s}\), it can be shown that \(E(Y) \approx h\left(\mu_{1}, \ldots, \mu_{n}\right)\) and $$ V(Y) \approx\left(\frac{\partial h}{\partial x_{1}}\right)^{2} \cdot \sigma_{1}^{2}+\cdots+\left(\frac{\partial h}{\partial x_{n}}\right)^{2} \cdot \sigma_{n}^{2} $$ where each partial derivative is evaluated at \(\left(x_{1}, \ldots, x_{n}\right)=\) \(\left(\mu_{1}, \ldots, \mu_{n}\right)\). Suppose three resistors with resistances \(X_{1}, X_{2}\), \(X_{3}\) are connected in parallel across a battery with voltage \(X_{4}\). Then by Ohm's law, the current is $$ Y=X_{4}\left[\frac{1}{X_{1}}+\frac{1}{X_{2}}+\frac{1}{X_{3}}\right] $$ Let \(\mu_{1}=10\) ohms, \(\sigma_{1}=1.0 \mathrm{ohm}, \mu_{2}=15 \mathrm{ohms}, \sigma_{2}=\) \(1.0 \mathrm{ohm}, \mu_{3}=20\) ohms, \(\sigma_{3}=1.5\) ohms, \(\mu_{4}=120 \mathrm{~V}\), \(\sigma_{4}=4.0 \mathrm{~V}\). Calculate the approximate expected value and standard deviation of the current (suggested by "Random Samplings," CHEMTECH, 1984: 696-697).

Step by step solution

01

Identify the function h

The given function for the current is \( Y = X_{4} \left[ \frac{1}{X_{1}} + \frac{1}{X_{2}} + \frac{1}{X_{3}} \right] \). Here, \( h(x_1, x_2, x_3, x_4) = x_4 \left[ \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} \right] \).

02

Calculate the approximate expected value of Y

Using the approximation for expected value, we have \( E(Y) \approx h(\mu_1, \mu_2, \mu_3, \mu_4) \). Substitute the given mean values: \( h(10, 15, 20, 120) = 120 \left[ \frac{1}{10} + \frac{1}{15} + \frac{1}{20} \right] \). Calculate this to get \( E(Y) \approx 120 \times 0.2167 = 26 \) A.

03

Compute partial derivatives of h

Find the partial derivatives \( \frac{\partial h}{\partial x_1} = -\frac{x_4}{x_1^2} \), \( \frac{\partial h}{\partial x_2} = -\frac{x_4}{x_2^2} \), \( \frac{\partial h}{\partial x_3} = -\frac{x_4}{x_3^2} \), and \( \frac{\partial h}{\partial x_4} = \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} \). Evaluate at \((x_1, x_2, x_3, x_4) = (10, 15, 20, 120)\).

04

Evaluate partial derivatives at mean values

Calculate \( \frac{\partial h}{\partial x_1} \text{ at } (10, 120) = -\frac{120}{10^2} = -1.2 \), \( \frac{\partial h}{\partial x_2} \text{ at } (15, 120) = -\frac{120}{15^2} = -0.5333 \), \( \frac{\partial h}{\partial x_3} \text{ at } (20, 120) = -\frac{120}{20^2} = -0.3 \), \( \frac{\partial h}{\partial x_4} \text{ at } (120) = 0.2167 \).

05

Calculate the approximate variance of Y

Use the formula \( V(Y) \approx \left(\frac{\partial h}{\partial x_1}\right)^2 \cdot \sigma_1^2 + \left(\frac{\partial h}{\partial x_2}\right)^2 \cdot \sigma_2^2 + \left(\frac{\partial h}{\partial x_3}\right)^2 \cdot \sigma_3^2 + \left(\frac{\partial h}{\partial x_4}\right)^2 \cdot \sigma_4^2 \). Substitute the values: \[ V(Y) \approx (1.2)^2 \times 1.0 + (0.5333)^2 \times 1.0 + (0.3)^2 \times 1.5 + (0.2167)^2 \times 4.0 \] Calculate this to get \( V(Y) = 2.137 \).

06

Determine the standard deviation of Y

The standard deviation \( SD(Y) \) is the square root of the variance: \( SD(Y) = \sqrt{2.137} \approx 1.461 \) A.

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Key Concepts

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

Random Variables

Random variables, often abbreviated as rv's, are fundamental elements in probability and statistics. They represent quantities whose outcomes depend on random phenomena. In the given exercise, the resistances of resistors in an electrical circuit are considered to be independent random variables denoted by \(X_1, X_2, X_3\), and they can take a range of values based on probability distributions.

Each of these random variables has a mean, represented by \(\mu\), which is the average value expected if we observed the variables numerous times. They also have a variance, denoted by \(\sigma^2\), which measures the dispersion of the random variable from the mean. These properties are crucial in calculating summaries, like the expected value and variance of derived functions such as \(Y\), which represents the current in the circuit using Ohm's Law.

• The mean \(\mu_i\) of each resistor indicates its average resistance value.
• The variance \(\sigma_i^2\) shows the variability of that resistance.

This approach allows us to make meaningful predictions and analyses about the system's behavior using probability theory.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering and physics, which states that the current \(Y\) flowing through a circuit is proportional to the voltage \(X_4\) and inversely proportional to the total resistance. In mathematical terms, it's expressed as \(Y = X_4 \left[ \frac{1}{X_1} + \frac{1}{X_2} + \frac{1}{X_3} \right]\), when the resistors are in parallel.

In the context of this problem, Ohm's Law is applied to determine the current through a set of resistors connected across a battery. The function \(h\) encapsulates this relationship by defining \(Y\) based on three parallel resistors, \(X_1, X_2,\) and \(X_3\), and their voltage \(X_4\).

• Parallel circuits reduce total resistance, making it less than the smallest individual resistor.
• The calculation involves reciprocal summation of resistances, hence the formula includes terms like \(\frac{1}{X_1}\).

Using this understanding of electrical current flow, we can predict and calculate properties such as the expected current and its variability.

Taylor Series Approximation

Taylor Series Approximations are used in mathematics to estimate complex functions as simpler polynomial forms. In this exercise, the main use of this concept is to approximate the expectation and variance of the current \(Y\), derived from the random variables \(X_1, X_2, X_3,\) and \(X_4\).

The Taylor Series is used here to handle the function \(h(X_1, X_2, X_3, X_4)\), approximating it around the mean values \(\mu_i\). This technique is useful when the variances \(\sigma_i^2\) are small relative to the means \(\mu_i\). It simplifies the computation of expected values, providing \(E(Y) \approx h(\mu_1, \ldots, \mu_n)\), and allows variance estimation through partial derivatives:

• Each partial derivative, such as \(\frac{\partial h}{\partial x_1}\), helps determine how sensitive \(Y\) is to changes in each \(X_i\).
• These derivatives are calculated at the known mean values, resulting in easier to manage expressions.

This approximation is a powerful tool in statistical analysis, especially when it involves calculating expectations and variances of complex functions.

Variance and Standard Deviation

Variance and standard deviation are key measures in statistics that describe the spread of values in a set of data. In the context of the exercise, once we determine the expected value of the current \(Y\) using the mean values, we move on to calculate how much the current might deviate from this average, which is described by variance \(V(Y)\) and standard deviation \(SD(Y)\).

Variance \( V(Y) \approx (\frac{\partial h}{\partial x_1})^2 \cdot \sigma_1^2 + (\frac{\partial h}{\partial x_2})^2 \cdot \sigma_2^2 + (\frac{\partial h}{\partial x_3})^2 \cdot \sigma_3^2 + (\frac{\partial h}{\partial x_4})^2 \cdot \sigma_4^2 \) takes into account how changes in each resistance and voltage affect \(Y\). Calculating each term involves squared partial derivatives multiplied by their respective variances. This shows the contribution of each variable's uncertainty to the overall uncertainty of \(Y\).

• Variance links directly to each variable's dispersion or spread.
• Taking the square root of the variance gives the standard deviation.

The standard deviation provides a more intuitive measure of spread by expressing it in the same units as the mean, allowing one to easily understand how much \(Y\)'s actual value may typically deviate from its expected value.