Let the correlation coefficient between X and Y be 0.6. Random variables Z and W are defined as `Z=X+5 and W=(Y)/(3)`. What is the correlation coefficient between Z and W?

Options

A. 0.1
B. 0.2
C. 0.36
D. 0.6

Explanation

To find the correlation coefficient between the random variables Z and W defined as \( Z = X + 5 \) and \( W = \frac{Y}{3} \), we can follow these steps:

### Step 1: Understand the relationship between Z, W, X, and Y
The correlation coefficient between two random variables X and Y is given as \( r_{XY} = 0.6 \). The random variables Z and W are transformations of X and Y:
- \( Z = X + 5 \) (adding a constant)
- \( W = \frac{Y}{3} \) (scaling)

### Step 2: Determine how transformations affect correlation
1. **Adding a constant**: The correlation coefficient is not affected by adding a constant to a variable. Therefore, the correlation between Z and X remains the same as the correlation between X and itself, which is 1.
2. **Scaling**: The correlation coefficient is affected by scaling. When a variable is multiplied by a constant, the correlation coefficient remains the same. Thus, the correlation between W and Y will also remain the same as the correlation between Y and itself, which is 1.

### Step 3: Calculate the correlation coefficient between Z and W
The correlation coefficient between Z and W can be calculated using the formula:
\[ r_{ZW} = \frac{\text{Cov}(Z, W)}{\sqrt{\text{Var}(Z) \cdot \text{Var}(W)}} \]

#### Step 3.1: Find the covariance of Z and W
Using the definitions of Z and W:
\[ \text{Cov}(Z, W) = \text{Cov}(X + 5, \frac{Y}{3}) = \text{Cov}(X, \frac{Y}{3}) = \frac{1}{3} \text{Cov}(X, Y) \]
Since \( \text{Cov}(X, Y) = r_{XY} \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)} \), we can express this as:
\[ \text{Cov}(Z, W) = \frac{1}{3} \cdot \text{Cov}(X, Y) \]

#### Step 3.2: Find the variances of Z and W
- For Z:
\[ \text{Var}(Z) = \text{Var}(X + 5) = \text{Var}(X) \quad \text{(adding a constant does not affect variance)} \]
- For W:
\[ \text{Var}(W) = \text{Var}\left(\frac{Y}{3}\right) = \left(\frac{1}{3}\right)^2 \text{Var}(Y) = \frac{1}{9} \text{Var}(Y) \]

### Step 4: Substitute into the correlation formula
Now substituting these into the correlation formula:
\[ r_{ZW} = \frac{\frac{1}{3} \text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \frac{1}{9} \text{Var}(Y)}} \]
This simplifies to:
\[ r_{ZW} = \frac{\frac{1}{3} \cdot r_{XY} \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}{\sqrt{\text{Var}(X) \cdot \frac{1}{9} \text{Var}(Y)}} \]
\[ = \frac{\frac{1}{3} \cdot r_{XY} \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}{\frac{1}{3} \sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]
\[ = r_{XY} \]

### Final Answer
Since \( r_{XY} = 0.6 \), we conclude:
\[ r_{ZW} = 0.6 \]

### Summary
The correlation coefficient between Z and W is \( 0.6 \).