If the correlation coefficient between x and y is 0.6, covariance is 27 and variance of y is 25, then what is the variance of x?

Options

A. `(9)/(5)`
B. `(81)/(25)`
C. 9
D. 81

Explanation

To find the variance of \( x \) given the correlation coefficient, covariance, and variance of \( y \), we can use the following relationship:

1. **Given Data**:
- Correlation coefficient \( r = 0.6 \)
- Covariance \( \text{Cov}(x, y) = 27 \)
- Variance of \( y \) \( \text{Var}(y) = 25 \)

2. **Formula for Correlation Coefficient**:
The correlation coefficient \( r \) can be expressed in terms of covariance and standard deviations:
\[ r = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y} \]
where \( \sigma_x \) is the standard deviation of \( x \) and \( \sigma_y \) is the standard deviation of \( y \).

3. **Calculate Standard Deviation of \( y \)**:
The standard deviation \( \sigma_y \) is the square root of the variance of \( y \):
\[ \sigma_y = \sqrt{\text{Var}(y)} = \sqrt{25} = 5 \]

4. **Substituting Values into the Correlation Formula**:
Now, substituting the known values into the correlation formula:
\[ 0.6 = \frac{27}{\sigma_x \cdot 5} \]

5. **Rearranging to Find \( \sigma_x \)**:
Rearranging the equation to solve for \( \sigma_x \):
\[ \sigma_x \cdot 5 = \frac{27}{0.6} \]
\[ \sigma_x = \frac{27}{0.6 \cdot 5} \]
\[ \sigma_x = \frac{27}{3} = 9 \]

6. **Calculating Variance of \( x \)**:
The variance of \( x \) is the square of the standard deviation:
\[ \text{Var}(x) = \sigma_x^2 = 9^2 = 81 \]

7. **Final Answer**:
Therefore, the variance of \( x \) is \( 81 \).