/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Let \(X_{1}\) and \(X_{2}\) have... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

Let \(X_{1}\) and \(X_{2}\) have a joint distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho\). Find the correlation coefficient of the linear functions of \(Y=a_{1} X_{1}+a_{2} X_{2}\) and \(Z=b_{1} X_{1}+b_{2} X_{2}\) in terms of the real constants \(a_{1}, a_{2}, b_{1}, b_{2}\), and the parameters of the distribution.

Short Answer

Expert verified
The correlation coefficient of the linear functions \(Y=a_{1} X_{1}+a_{2} X_{2}\) and \(Z=b_{1} X_{1}+b_{2} X_{2}\) is given by the equation \(\rho_{YZ} = \frac{a_{1}b_{1}\sigma_{1}^{2} + a_{2}b_{2}\sigma_{2}^{2} + \rho a_{1}b_{2}\sigma_{1}\sigma_{2} + \rho a_{2}b_{1}\sigma_{1}\sigma_{2}}{\sqrt{(a_{1}^{2}\sigma_{1}^{2} + a_{2}^{2}\sigma_{2}^{2} + 2a_{1}a_{2}\rho\sigma_{1}\sigma_{2})(b_{1}^{2}\sigma_{1}^{2} + b_{2}^{2}\sigma_{2}^{2} + 2b_{1}b_{2}\rho\sigma_{1}\sigma_{2})}}\).

Step by step solution

01

Compute Covariance

First, we compute the covariance of Y and Z. Using the formula for covariance, we get \(Cov(Y, Z) = E[(Y-E(Y))(Z-E(Z))] = E[YZ] - E(Y)E(Z)\). Simplifying, we get \(Cov(Y, Z) = a_{1}b_{1}\sigma_{1}^{2} + a_{2}b_{2}\sigma_{2}^{2} + \rho a_{1}b_{2}\sigma_{1}\sigma_{2} + \rho a_{2}b_{1}\sigma_{1}\sigma_{2}\).
02

Compute Variance of Y and Z

We then compute the variance of Y and Z. Using the formula for variance, we get \(Var(Y) = E[(Y-E(Y))^{2}]\) and \(Var(Z) = E[(Z-E(Z))^{2}]\). Simplifying, we get \(Var(Y) = a_{1}^{2}\sigma_{1}^{2} + a_{2}^{2}\sigma_{2}^{2} + 2a_{1}a_{2}\rho\sigma_{1}\sigma_{2}\) for Y, and \(Var(Z) = b_{1}^{2}\sigma_{1}^{2} + b_{2}^{2}\sigma_{2}^{2} + 2b_{1}b_{2}\rho\sigma_{1}\sigma_{2}\) for Z.
03

Compute Correlation Coefficient

Finally, we plug our results into our correlation coefficient formula, \(\rho_{YZ} = \frac{Cov(Y,Z)}{\sqrt{Var(Y)Var(Z)}}\). After substitution, our final formula becomes: \(\rho_{YZ} = \frac{a_{1}b_{1}\sigma_{1}^{2} + a_{2}b_{2}\sigma_{2}^{2} + \rho a_{1}b_{2}\sigma_{1}\sigma_{2} + \rho a_{2}b_{1}\sigma_{1}\sigma_{2}}{\sqrt{(a_{1}^{2}\sigma_{1}^{2} + a_{2}^{2}\sigma_{2}^{2} + 2a_{1}a_{2}\rho\sigma_{1}\sigma_{2})(b_{1}^{2}\sigma_{1}^{2} + b_{2}^{2}\sigma_{2}^{2} + 2b_{1}b_{2}\rho\sigma_{1}\sigma_{2})}}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers. Hence, we can also say that \(\left\\{a_{n}\right\\}\) is a sequence of constant (degenerate) random variables. Let \(a\) be a real number. Show that \(a_{n} \rightarrow a\) is equivalent to \(a_{n} \stackrel{P}{\rightarrow} a\)

Forty-eight measurements are recorded to several decimal places. Each of these 48 numbers is rounded off to the nearest integer. The sum of the original 48 numbers is approximated by the sum of these integers. If we assume that the errors made by rounding off are iid and have a uniform distribution over the interval \(\left(-\frac{1}{2}, \frac{1}{2}\right)\), compute approximately the probability that the sum of the integers is within two units of the true sum.

For the last exercise, suppose that the sample is drawn from a \(N\left(\mu, \sigma^{2}\right)\) distribution. Recall that \((n-1) S^{2} / \sigma^{2}\) has a \(\chi^{2}(n-1)\) distribution. Use Theorem 3.3.1 to determine an unbiased estimator of \(\sigma\).

Let \(Y_{n}\) denote the maximum of a random sample from a distribution of the continuous type that has cdf \(F(x)\) and pdf \(f(x)=F^{\prime}(x)\). Find the limiting distribution of \(Z_{n}=n\left[1-F\left(Y_{n}\right)\right]\)

Let \(X\) be \(N\left(\mu, \sigma^{2}\right)\) and consider the transformation \(X=\log (Y)\) or, equivalently, \(Y=e^{X}\) (a) Find the mean and the variance of \(Y\) by first determining \(E\left(e^{X}\right)\) and \(E\left[\left(e^{X}\right)^{2}\right]\), by using the mgf of \(X\). (b) Find the pdf of \(Y\). This is the pdf of the lognormal distribution.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.