/*! 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 14 Let \(X_{1}\) and \(X_{2}\) have... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 14

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
\[\rho_{YZ} = \frac{a_1b_1\sigma_1^2 + a_2b_2\sigma_2^2 + (a_1b_2 + a_2b_1)\rho\sigma_1\sigma_2}{\sqrt{(a_1^2\sigma_1^2 + a_2^2\sigma_2^2 + 2a_1a_2\rho\sigma_1\sigma_2)(b_1^2\sigma_1^2 + b_2^2\sigma_2^2 + 2b_1b_2\rho\sigma_1\sigma_2)}}\]

Step by step solution

01

Finding statistical expectations

First, find the expected values of \(Y\) and \(Z\) and their variances. The expected value \(E(Y)\) is \(a_1\mu_1 + a_2\mu_2\) and \(E(Z)\) is \(b_1\mu_1 + b_2\mu_2\). The variances are: \(Var(Y) = a_1^2\sigma_1^2 + a_2^2\sigma_2^2 + 2a_1a_2\rho\sigma_1\sigma_2\) and \(Var(Z) = b_1^2\sigma_1^2 + b_2^2\sigma_2^2 + 2b_1b_2\rho\sigma_1\sigma_2\).
02

Finding covariance

Next is to calculate the covariance of \(Y\) and \(Z\), given by \(Cov(Y,Z) = a_1b_1\sigma_1^2 + a_2b_2\sigma_2^2 + (a_1b_2 + a_2b_1)\rho\sigma_1\sigma_2\).
03

Finding the correlation coefficient

Finally, compute the correlation coefficient of \(Y\) and \(Z\) which is defined by \[\rho_{YZ} = \frac{Cov(Y, Z)}{\sqrt{Var(Y)Var(Z)}}\] Substituting the calculated values of expectations, variances, and covariance into this formula gives the correlation coefficient in terms of the provided real constants and the joint distribution parameters.

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