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Chapter 5: Problem 95

Let \(X\) and \(Y\) be independent standard normal random variables, and define a new rv by \(U=.6 X+.8 Y\). a. Determine \(\operatorname{Corr}(X, U)\). b. How would you alter \(U\) to obtain \(\operatorname{Corr}(X, U)=\rho\) for a specified value of \(\rho\) ?

Short Answer

Expert verified
a. \(\operatorname{Corr}(X, U) = 0.6\). b. Adjust the coefficient of \(X\) in \(U\) to get \(a' = \rho\times\sqrt{1 + b^2/a^2}\).

Step by step solution

01

Understand the given variables

We have two independent standard normal random variables, namely, \(X\) and \(Y\). Both have a mean of 0 and a variance of 1. The newly defined random variable \(U\) is expressed as a linear combination of \(X\) and \(Y\): \(U = 0.6X + 0.8Y\).
02

Identify the coefficients in U

The expression for \(U\) shows that it is composed of two parts: \(0.6X\) and \(0.8Y\). Thus, the coefficient of \(X\) is 0.6 and the coefficient of \(Y\) is 0.8.
03

Apply the formula for correlation

Since \(X\) and \(Y\) are independent, and consequently have a correlation of 0, the correlation between \(X\) and \(U\) can be calculated using the formula for correlation of linear combinations: \(\operatorname{Corr}(X, aX + bY) = \frac{a}{\sqrt{a^2 + b^2}}\).
04

Calculate the correlation between X and U

Substitute the coefficients in the correlation formula: \[\operatorname{Corr}(X, U) = \frac{0.6}{\sqrt{0.6^2 + 0.8^2}}\].Calculate the squares: \(0.6^2 = 0.36\) and \(0.8^2 = 0.64\). Add them to get \(1\): \(0.36 + 0.64 = 1\). The square root of 1 is 1, thus \[\operatorname{Corr}(X, U) = \frac{0.6}{1} = 0.6\].
05

Alter U to achieve specified correlation

To alter \(U\) so that \(\operatorname{Corr}(X, U) = \rho\), we use the correlation formula to express \(a\) in terms of \(\rho\) and \(b\): \[\rho = \frac{a}{\sqrt{a^2 + b^2}}\].Rearranging this equation, solve for \(a\): \[a = \rho \times \sqrt{a^2 + b^2}\].Let us fix \(a' = \rho\times\sqrt{\frac{1}{a}\cdot b^2 + 1}\) to create a desired correlation with \(b\) remaining unchanged.

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Key Concepts

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

Linear Combinations
A linear combination of random variables involves taking two or more random variables and combining them using specific coefficients. In simple terms, you multiply each random variable by a coefficient and sum the results. In the exercise, the random variable \( U \) is expressed as \( U = 0.6X + 0.8Y \), indicating that \( U \) is formed by scaling the variable \( X \) by 0.6 and \( Y \) by 0.8, then adding them together.

This concept is crucial in statistics because it allows us to create new random variables based on existing ones, while potentially changing their properties such as mean and variance.
You can:
  • Adjust how much each original random variable affects the outcome by changing the coefficients.
  • Analyze complex datasets by interpreting them as combinations of simpler, well-understood random variables.
The power of linear combinations lies in this ability to create flexibility and modify statistical properties, helping in fields like risk management and financial modeling.
Standard Normal Distribution
A standard normal distribution is a special type of normal distribution.
It has a mean of 0 and a variance of 1, traditionally denoted as \( N(0, 1) \). Both \( X \) and \( Y \) from the exercise are independent standard normal random variables. Independence means the occurrence of one does not affect the occurrence of the other.

The properties of the standard normal distribution are often used in statistics for various applications:
  • It serves as the basis for the z-score, which is a measure of how many standard deviations an element is from the mean.
  • It is foundational for various statistical tests, including those used to infer the mean of a data set.
The simplicity and symmetry of the standard normal distribution make it a cornerstone in statistical theory and practice, aiding in the calculation of probabilities and prediction of outcomes.
Correlation Coefficient
The correlation coefficient, sometimes referred to as Pearson's correlation, is a measure that determines the degree to which two random variables move in relation to each other.

It is expressed as a value between -1 and 1:
  • A value of 1 implies a perfect positive correlation, where variables move in the same direction.
  • A value of -1 implies a perfect negative correlation, where variables move in opposite directions.
  • A value of 0 indicates no correlation, suggesting no discernible relationship in movement between the variables.
In the context of this exercise, the formula \( \operatorname{Corr}(X, aX + bY) = \frac{a}{\sqrt{a^2 + b^2}} \) is used to calculate the correlation of \( X \) and \( U \).
This method allows us to understand how one component of a linear combination relates to the original variable. Adjusting \( U \) to alter the correlation, as shown in the solution, involves carefully choosing coefficients to reach a desired correlation value. This technique is widely used in fields such as finance and research to refine models and achieve specific statistical properties.

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Most popular questions from this chapter

Two airplanes are flying in the same direction in adjacent parallel corridors. At time \(t=0\), the first airplane is \(10 \mathrm{~km}\) ahead of the second one. Suppose the speed of the first plane \((\mathrm{km} / \mathrm{hr})\) is normally distributed with mean 520 and standard deviation 10 and the second plane's speed is also normally distributed with mean and standard deviation 500 and 10 , respectively. a. What is the probability that after \(2 \mathrm{hr}\) of flying, the second plane has not caught up to the first plane? b. Determine the probability that the planes are separated by at most \(10 \mathrm{~km}\) after \(2 \mathrm{hr}\).

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. b. Would your calculations necessarily be correct if the \(X_{i} \mathrm{~s}\) were not independent? Explain.

Let \(X\) denote the number of Canon digital cameras sold during a particular week by a certain store. The pmf of \(X\) is $$ \begin{array}{l|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline p_{X}(x) & .1 & .2 & .3 & .25 & .15 \end{array} $$ Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let \(Y\) denote the number of purchasers during this week who buy an extended warranty. a. What is \(P(X=4, Y=2)\) ? [Hint: This probability equals \(P(Y=2 \mid X=4) \cdot P(X=4)\); now think of the four purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended warranty.] b. Calculate \(P(X=Y)\). c. Determine the joint pmf of \(X\) and \(Y\) and then the marginal pmf of \(Y\).

Six individuals, including \(\mathrm{A}\) and \(\mathrm{B}\), take seats around a circular table in a completely random fashion. Suppose the seats are numbered \(1, \ldots, 6\). Let \(X=\) A's seat number and \(Y=\) B's seat number. If A sends a written message around the table to \(\mathrm{B}\) in the direction in which they are closest, how many individuals (including \(\mathrm{A}\) and \(\mathrm{B}\) ) would you expect to handle the message?

Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the table. $$ \begin{array}{l|ccc} & \text { Road 1 } & \text { Road 2 } & \text { Road 3 } \\ \hline \text { Expected value } & 800 & 1000 & 600 \\ \text { Standard deviation } & 16 & 25 & 18 \end{array} $$ a. What is the expected total number of cars entering the freeway at this point during the period? b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the numbers of cars on the different roads? c. With \(X_{i}\) denoting the number of cars entering from road \(i\) during the period, suppose that \(\operatorname{Cov}\left(X_{1}, X_{2}\right)=80\), \(\operatorname{Cov}\left(X_{1}, X_{3}\right)=90\), and \(\operatorname{Cov}\left(X_{2}, X_{3}\right)=100\) (so that the three streams of traffic are not independent). Compute the expected total number of entering cars and the standard deviation of the total.
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