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Chapter 2: Problem 10

Determine the correlation coefficient of the random variables \(X\) and \(Y\) if \(\operatorname{var}(X)=4, \operatorname{var}(Y)=2\), and \(\operatorname{var}(X+2 Y)=15\)

Short Answer

Expert verified
The correlation coefficient of the random variables X and Y is \(0.53\).

Step by step solution

01

Apply the formula for the variance

Applying the formula for the variance of the sum of two random variables, \(\operatorname{var}(X+2Y)=\operatorname{var}(X)+4*\operatorname{var}(Y)+4*\operatorname{cov}(X,Y)\). Substitute the given values for \(\operatorname{var}(X)\), \(\operatorname{var}(Y)\), and \(\operatorname{var}(X+2Y)\) in this formula to get an equation that can be solved for \(\operatorname{cov}(X,Y)\).
02

Solve for the covariance

Solve the equation for \(\operatorname{cov}(X,Y)\) to find its value. Use the equation \(15=4+4*2+4*\operatorname{cov}(X,Y)\) to obtain \(\operatorname{cov}(X,Y)\) as \(0.75\).
03

Compute the correlation coefficient

The formula for the correlation coefficient is \(\rho(X,Y)=\frac{\operatorname{cov}(X,Y)}{\sqrt{\operatorname{var}(X)*\operatorname{var}(Y)}}\). Substitute the values of \(\operatorname{cov}(X,Y)\), \(\operatorname{var}(X)\), and \(\operatorname{var}(Y)\) in this formula to solve for \(\rho(X,Y)\).
04

Final computation

The substitution of the values in the formula gives \(\rho(X,Y)=\frac{0.75}{\sqrt{4*2}}\). Continuing with the computation will give you the final answer for the correlation coefficient.

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