Exercise Correlation properties 4 6 points graded As in the preceding example let Z V and W be independent random variables with mean 0 and variance 1 and let X Z V and Y Z W We have found that X Y 1...

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Exercise: Correlation properties 4/6 points (graded) As in the preceding example, let $Z, V$, and $W$ be independent random variables with mean 0 and variance 1 , and let $X=Z+V$ and $Y=Z+W$. We have found that $\rho(X, Y)=1 / 2$. a) It follows that: \[ \rho(X,-Y)=-1 / 2 \] \[ \rho(-X,-Y)=1 / 2 \] b) Suppose that $X$ and $Y$ are measured in dollars. Let $X^{\prime}$ and $Y^{\prime}$ be the same random variables, but measured in cents, so that $X^{\prime}=100 X$ and $Y^{\prime}=100 Y$. Then, \[ \rho\left(X^{\prime}, Y^{\prime}\right)=1 / 2 \] c) Suppose now that $\tilde{X}=3 Z+3 V+3$ and $\tilde{Y}=-2 Z-2 W$. Then \[ \rho(\tilde{X}, \tilde{Y})=-1 / 2 \] d) Suppose now that the variance of $Z$ is replaced by a very large number. Then $\rho(X, Y)$ is close to e) Alternatively, suppose that the variance of $Z$ is close to zero. Then $\rho(X, Y)$ is close to

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#### Solution By Steps ***Step 1: Calculate $\rho(X, Y)$*** Given that $\rho(X, Y) = 1/2$, this implies that the correlation coefficient between $X$ and $Y$ is $1/2$. ***Step 2: Calculate $\rho(X, -Y)$*** Since $X = Z + V$ and $Y = Z + W$, we have $-Y = -Z - W$. The correlation coefficient between $X$ and $-Y$ is the negative of the correlation coefficient between $X$ and $Y$. Therefore, $\rho(X, -Y) = -1/2$. ***Step 3: Calculate $\rho(-X, -Y)$*** Similarly, $-X = -Z - V$. The correlation coefficient between $-X$ and $-Y$ is the same as the correlation coefficient between $X$ and $Y$, which is $1/2$. Hence, $\rho(-X, -Y) = 1/2$. ***Step 4: Calculate $\rho(X', Y')$*** Given $X' = 100X$ and $Y' = 100Y$, the correlation coefficient between $X'$ and $Y'$ remains the same as the correlation coefficient between $X$ and $Y$, which is $1/2$. ***Step 5: Calculate $\rho(\tilde{X}, \tilde{Y})$*** For $\tilde{X} = 3Z + 3V + 3$ and $\tilde{Y} = -2Z - 2W$, the correlation coefficient between $\tilde{X}$ and $\tilde{Y}$ is $-1/2$. ***Step 6: Determine the effect of changing the variance of $Z$*** When the variance of $Z$ becomes very large, the correlation coefficient $\rho(X, Y)$ approaches 1. ***Step 7: Determine the effect of a small variance of $Z$*** If the variance of $Z$ is close to zero, the correlation coefficient $\rho(X, Y)$ tends towards 0. #### Final Answer a) $\rho(X, -Y) = -1/2$, $\rho(-X, -Y) = 1/2$ b) $\rho(X', Y') = 1/2$ c) $\rho(\tilde{X}, \tilde{Y}) = -1/2$ d) $\rho(X, Y)$ is close to 1 e) $\rho(X, Y)$ is close to 0 #### Key Concept Correlation Coefficient #### Key Concept Explanation The correlation coefficient measures the strength and direction of a linear relationship between two random variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. Changes in variances or scaling of variables can affect the correlation coefficient.

Follow-up Knowledge or Question

What is the correlation between $X$ and $-Y$ if $\rho(X, Y) = 1/2$?

How does changing the units of measurement affect the correlation between random variables?

How does changing the coefficients in the linear combinations of random variables affect their correlation?

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