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

Options

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

Answer

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

\[ r_{xy} = \frac{\text{Cov}(x, y)}{\sqrt{\text{Var}(x) \cdot \text{Var}(y)}} \]

Where:
- \( r_{xy} \) is the coefficient of correlation between \( x \) and \( y \),
- \( \text{Cov}(x, y) \) is the covariance between \( x \) and \( y \),
- \( \text{Var}(x) \) is the variance of \( x \),
- \( \text{Var}(y) \) is the variance of \( y \).

Given:
- \( r_{xy} = 0.6 \)
- \( \text{Cov}(x, y) = 27 \)
- \( \text{Var}(y) = 25 \)

We need to find \( \text{Var}(x) \).

### Step 1: Substitute the known values into the formula

Substituting the known values into the correlation formula:

\[ 0.6 = \frac{27}{\sqrt{\text{Var}(x) \cdot 25}} \]

### Step 2: Rearrange the equation

To isolate \( \sqrt{\text{Var}(x)} \), we can multiply both sides by \( \sqrt{\text{Var}(x) \cdot 25} \):

\[ 0.6 \cdot \sqrt{\text{Var}(x) \cdot 25} = 27 \]

Next, divide both sides by \( 0.6 \):

\[ \sqrt{\text{Var}(x) \cdot 25} = \frac{27}{0.6} \]

Calculating the right side:

\[ \frac{27}{0.6} = 45 \]

So we have:

\[ \sqrt{\text{Var}(x) \cdot 25} = 45 \]

### Step 3: Square both sides

Now, square both sides to eliminate the square root:

\[ \text{Var}(x) \cdot 25 = 45^2 \]

Calculating \( 45^2 \):

\[ 45^2 = 2025 \]

So we have:

\[ \text{Var}(x) \cdot 25 = 2025 \]

### Step 4: Solve for \( \text{Var}(x) \)

Now, divide both sides by 25 to find \( \text{Var}(x) \):

\[ \text{Var}(x) = \frac{2025}{25} \]

Calculating the division:

\[ \text{Var}(x) = 81 \]

### Final Answer

Thus, the variance of \( x \) is:

\[ \text{Var}(x) = 81 \]