If coefficient of correlation is -0.6, standard deviation of x is 5 and variance of y is 16, then what is the co-variance of x and y?

Options

A. 0.75
B. 0.48
C. -0.03
D. -0.75

Answer

To find the covariance of x and y given the coefficient of correlation, standard deviation of x, and variance of y, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between correlation, covariance, and standard deviations**:
The formula relating these quantities is:
\[ R = \frac{\text{Cov}(x, y)}{\sigma_x \cdot \sigma_y} \]
where \( R \) is the coefficient of correlation, \( \text{Cov}(x, y) \) is the covariance of x and y, \( \sigma_x \) is the standard deviation of x, and \( \sigma_y \) is the standard deviation of y.

2. **Identify the given values**:
- Coefficient of correlation \( R = -0.6 \)
- Standard deviation of x \( \sigma_x = 5 \)
- Variance of y \( \text{Var}(y) = 16 \)

3. **Calculate the standard deviation of y**:
Since variance is the square of the standard deviation, we can find \( \sigma_y \) as follows:
\[ \sigma_y = \sqrt{\text{Var}(y)} = \sqrt{16} = 4 \]

4. **Substitute the known values into the correlation formula**:
Now we can rearrange the formula to solve for covariance:
\[ \text{Cov}(x, y) = R \cdot \sigma_x \cdot \sigma_y \]
Substituting the known values:
\[ \text{Cov}(x, y) = -0.6 \cdot 5 \cdot 4 \]

5. **Calculate the covariance**:
\[ \text{Cov}(x, y) = -0.6 \cdot 20 = -12 \]

6. **Final answer**:
Therefore, the covariance of x and y is:
\[ \text{Cov}(x, y) = -12 \]