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Chapter 5: Problem 32

a. Use the general formula for the variance of a linear combination to write an expression for \(V(a X+Y)\). Then let \(a=\sigma_{Y} / \sigma_{X}\), and show that \(\rho \geq-1\). [Hint: Variance is always \(\geq 0\), and \(\left.\operatorname{Cov}(X, Y)=\sigma_{X} \cdot \sigma_{Y} \cdot \rho .\right]\) b. By considering \(V(a X-Y)\), conclude that \(\rho \leq 1\). c. Use the fact that \(V(W)=0\) only if \(W\) is a constant to show that \(\rho=1\) only if \(Y=a X+b\).

Short Answer

Expert verified
The correlation \(\rho\) is between -1 and 1, and \(\rho = 1\) implies \(Y = aX + b\).

Step by step solution

01

General Formula for Variance of Linear Combination

The variance of a linear combination of random variables, such as \(aX + Y\), is given by:\[V(aX + Y) = a^2V(X) + V(Y) + 2a \cdot \text{Cov}(X, Y) \]Using this, plug in the values for \(V(X) = \sigma_X^2\), \(V(Y) = \sigma_Y^2\), and \(\text{Cov}(X, Y) = \sigma_X\sigma_Y\rho\) into the formula.
02

Substitution of Specific Value for a

Given \(a = \frac{\sigma_Y}{\sigma_X}\), substitute into the variance expression:\[V\left(\frac{\sigma_Y}{\sigma_X}X + Y\right) = \left(\frac{\sigma_Y}{\sigma_X}\right)^2 \sigma_X^2 + \sigma_Y^2 + 2 \frac{\sigma_Y}{\sigma_X} \sigma_X \sigma_Y \rho\]This simplifies to:\[V\left(\frac{\sigma_Y}{\sigma_X}X + Y\right) = \sigma_Y^2 + \sigma_Y^2 + 2\sigma_Y^2 \rho\]
03

Apply Variance Non-Negativity and Solve for \(\rho\)

Since variance is always non-negative, we have:\[\sigma_Y^2(1 + 1 + 2\rho) \geq 0\]\[2\sigma_Y^2(1 + \rho) \geq 0\]Since \(\sigma_Y^2 > 0\), divide both sides by \(2\sigma_Y^2\), giving:\[1 + \rho \geq 0\]Thus, \(\rho \geq -1\).
04

Consider Variance of \(aX - Y\) for Upper Bound

Now use the variance of \(aX - Y\):\[V(aX - Y) = a^2V(X) + V(Y) - 2a \cdot \text{Cov}(X, Y)\]Substitute \(a = \frac{\sigma_Y}{\sigma_X}\) and simplify:\[\sigma_Y^2(1 - \rho) \geq 0 \]Since \(\sigma_Y^2 > 0\), divide both sides by \(\sigma_Y^2\), yielding:\[1 - \rho \geq 0\]Thus, \(\rho \leq 1\).
05

Equality Condition for \(\rho = 1\)

If \(\rho = 1\), variance calculation implies \(V\left(\frac{\sigma_Y}{\sigma_X}X - Y\right) = 0\).The variance of any random variable is zero only if it is a constant. Thus, \(\frac{\sigma_Y}{\sigma_X}X - Y = c\) (a constant), implying \(Y = \frac{\sigma_Y}{\sigma_X}X + b\), where \(b\) is some constant.

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Key Concepts

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

Covariance
Covariance is a fundamental concept in statistics and probability that measures the extent to which two random variables change together. If we have two random variables, say, \(X\) and \(Y\), their covariance is defined as:\[\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]\]where \(E\) denotes the expected value. The covariance can be positive, negative, or zero.
  • Positive covariance: When \(X\) tends to be above its mean, \(Y\) is also above its mean.
  • Negative covariance: When \(X\) tends to be above its mean, \(Y\) is below its mean.
  • Zero covariance: The variables are uncorrelated, without any linear relationship.
Covariance is a crucial part of calculating the variance of linear combinations. For example, in a linear combination like \(aX + Y\), the variance incorporates the covariance between \(X\) and \(Y\) to account for how they vary together.
Correlation Coefficient
The correlation coefficient, often symbolized by \(\rho\), is a standardized version of covariance. It measures the strength and direction of a linear relationship between two random variables. The correlation coefficient is defined as:\[\rho_{XY} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}\]where \(\sigma_X\) and \(\sigma_Y\) are the standard deviations of \(X\) and \(Y\), respectively.
The value of \(\rho\) is always between -1 and 1. This restriction helps in understanding the degree of the linear relationship:
  • \(\rho = 1\): Perfect positive correlation. \(Y\) increases as \(X\) increases.
  • \(\rho = -1\): Perfect negative correlation. \(Y\) decreases as \(X\) increases.
  • \(\rho = 0\): No linear correlation. \(X\) and \(Y\) move independently of each other.
The correlation coefficient is dimensionless, making it a versatile tool for comparing relationships across different datasets and units.
Random Variables
Random variables are fundamental in probability and statistics, representing quantities with uncertain outcomes. A random variable can take on various values, each with a probability assigned to it. There are two main types of random variables:
  • Discrete random variables: These take on a countable number of distinct values. For example, the roll of a die can result in any integer from 1 to 6.
  • Continuous random variables: These can take on an infinite number of possible values within a given range. For example, the exact time taken to run a marathon.
Each random variable has a probability distribution that describes the probabilities of different outcomes. Variance and covariance are key properties related to random variables that describe their spread and interdependencies.
The understanding of random variables is crucial for deriving equations related to variance, covariance, and correlation, particularly when dealing with linear combinations as shown in the exercise.

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Most popular questions from this chapter

Let \(X_{1}, \ldots, X_{n}\) be independent rv's with mean values \(\mu_{1}, \ldots, \mu_{n}\) and variances \(\sigma_{1}^{2}, \ldots, \sigma_{\omega}^{2}\). Consider a function \(h\left(x_{1}, \ldots, x_{n}\right)\), and use it to define a rv \(Y=h\left(X_{1}, \ldots, X_{n}\right)\). Under rather general conditions on the \(h\) function, if the \(\sigma_{i}^{\prime}\) 's are all small relative to the corresponding \(\mu_{i}\) 's, it can be shown that \(E(Y) \approx h\left(\mu_{1}, \ldots, \mu_{n}\right)\) and $$ V(Y) \propto\left(\frac{\partial h}{\partial x_{1}}\right)^{2} \cdot \sigma_{1}^{2}+\cdots+\left(\frac{\partial h}{\partial x_{n}}\right)^{2} \cdot \sigma_{n}^{2} $$ 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, \(\sigma_{1}=1.0 \mathrm{ohm}, \quad \mu_{2}=15\) ohms, \(\sigma_{2}=1.0 \mathrm{ohm}, \mu_{3}=20\) ohms, \(\sigma_{3}=1.5 \mathrm{ohms}, \mu_{4}=120 \mathrm{~V}\), \(\sigma_{4}=4.0 \mathrm{~V}\). Calculate the approximate expected value and standard deviation of the current (suggested by "Random Samplings," CHEMTECH, 1984: 696-697).

A rock specimen from a particular area is randomly selected and weighed two different times. Let \(W\) denote the actual weight and \(X_{1}\) and \(X_{2}\) the two measured weights. Then \(X_{1}=W+E_{1}\) and \(X_{2}=W+E_{2}\), where \(E_{1}\) and \(E_{2}\) are the two measurement errors. Suppose that the \(E_{i}\) 's are independent of one another and of \(W\) and that \(V\left(E_{1}\right)=V\left(E_{2}\right)=\sigma_{E}^{2}\). a. Express \(\rho\), the correlation coefficient between the two measured weights \(X_{1}\) and \(X_{2}\), in terms of \(\sigma_{W}^{2}\), the variance of actual weight, and \(\sigma_{X}^{2}\), the variance of measured weight. b. Compute \(\rho\) when \(\sigma_{W}=1 \mathrm{~kg}\) and \(\sigma_{E}=.01 \mathrm{~kg}\).

A shipping company handles containers in three different sizes: (1) \(27 \mathrm{ft}^{3}(3 \times 3 \times 3)\), (2) \(125 \mathrm{ft}^{3}\), and (3) \(512 \mathrm{ft}^{3}\). Let \(X_{i}(i=1,2,3)\) denote the number of type \(i\) containers shipped during a given week. With \(\mu_{i}=E\left(X_{i}\right)\) and \(\sigma_{i}^{2}=V\left(X_{i}\right)\), suppose that the mean values and standard deviations are as follows: $$ \begin{array}{lll} \mu_{1}=200 & \mu_{2}=250 & \mu_{3}=100 \\ \sigma_{1}=10 & \sigma_{2}=12 & \sigma_{3}=8 \end{array} $$ a. Assuming that \(X_{1}, X_{2}, X_{3}\) are independent, calculate the expected value and variance of the total volume shipped. [Hint: Volume \(=27 X_{1}+125 X_{2}+512 X_{3}\).] b. Would your calculations necessarily be correct if the \(X_{i}^{\prime}\) s were not independent? Explain.

Suppose the expected tensile strength of type-A steel is \(105 \mathrm{ksi}\) and the standard deviation of tensile strength is \(8 \mathrm{ksi}\). For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are \(100 \mathrm{ksi}\) and \(6 \mathrm{ksi}\), respectively. Let \(\bar{X}=\) the sample average tensile strength of a random sample of 40 type-A specimens, and let \(\bar{Y}=\) the sample average tensile strength of a random sample of 35 type-B specimens. a. What is the approximate distribution of \(\bar{X}\) ? Of \(\bar{Y}\) ? b. What is the approximate distribution of \(\bar{X}-\bar{Y}\) ? Justify your answer. c. Calculate (approximately) \(P(-1 \leq \bar{X}-\bar{Y} \leq 1)\). d. Calculate \(P(\bar{X}-\bar{Y} \geq 10)\). If you actually observed \(\bar{X}-\bar{Y} \geq 10\), would you doubt that \(\mu_{1}-\mu_{2}=5\) ?

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of \(\bar{X}\) when the population distribution is lognormal with \(E(\ln (X))=3\) and \(V(\ln (X))=1\). Consider the four sample sizes \(n=10,20,30\), and 50 , and in each case use 1000 replications. For which of these sample sizes does the \(\bar{X}\) sampling distribution appear to be approximately normal?
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