The coefficient of correlation between random variables X and Y is 0.28, covariance between X and Y is 7.6 and the variance of X is 9, then the standard deviation of Y series is

Options

A. `9.8`
B. `10.1`
C. `9.05`
D. `10.05`

Answer

To find the standard deviation of the Y series given the correlation coefficient, covariance, and variance of X, we can use the formula for the correlation coefficient (r):

### Step-by-Step Solution:

1. **Understand the formula for the correlation coefficient (r)**:
\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} \]
where:
- \( r \) is the coefficient of correlation,
- \( \text{Cov}(X, Y) \) is the covariance between X and Y,
- \( \sigma_X \) is the standard deviation of X,
- \( \sigma_Y \) is the standard deviation of Y.

2. **Identify the given values**:
- Coefficient of correlation, \( r = 0.28 \)
- Covariance, \( \text{Cov}(X, Y) = 7.6 \)
- Variance of X, \( \text{Var}(X) = 9 \)

3. **Calculate the standard deviation of X**:
\[ \sigma_X = \sqrt{\text{Var}(X)} = \sqrt{9} = 3 \]

4. **Rearrange the correlation formula to solve for \( \sigma_Y \)**:
\[ \sigma_Y = \frac{\text{Cov}(X, Y)}{r \cdot \sigma_X} \]

5. **Substitute the known values into the rearranged formula**:
\[ \sigma_Y = \frac{7.6}{0.28 \cdot 3} \]

6. **Calculate the denominator**:
\[ 0.28 \cdot 3 = 0.84 \]

7. **Now calculate \( \sigma_Y \)**:
\[ \sigma_Y = \frac{7.6}{0.84} \approx 9.05 \]

### Final Answer:
The standard deviation of the Y series is approximately **9.05**.

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