The coefficient of correlation between X and Y is 0.6. Their covariance is 4.8 var (x)=9. Find the value of `sigma_(y)`.

Answer

To find the value of the standard deviation of Y (denoted as σ_y), we can use the relationship between the coefficient of correlation (r), covariance (Cov(X,Y)), and the standard deviations of X and Y. The formula for the coefficient of correlation is given by:

\[ r = \frac{Cov(X,Y)}{\sigma_x \cdot \sigma_y} \]

Where:
- \( r \) is the coefficient of correlation,
- \( Cov(X,Y) \) is the covariance between X and Y,
- \( \sigma_x \) is the standard deviation of X,
- \( \sigma_y \) is the standard deviation of Y.

Given in the question:
- \( r = 0.6 \)
- \( Cov(X,Y) = 4.8 \)
- \( Var(X) = 9 \)

Step 1: Calculate the standard deviation of X (σ_x).
Since the variance of X is given as 9, we can find the standard deviation by taking the square root of the variance:

\[ \sigma_x = \sqrt{Var(X)} = \sqrt{9} = 3 \]

Step 2: Substitute the known values into the correlation formula.
Now we can substitute the known values into the correlation formula to find σ_y:

\[ 0.6 = \frac{4.8}{3 \cdot \sigma_y} \]

Step 3: Rearranging the equation to solve for σ_y.
To isolate σ_y, we can rearrange the equation:

\[ 0.6 \cdot 3 \cdot \sigma_y = 4.8 \]

\[ 1.8 \cdot \sigma_y = 4.8 \]

Step 4: Solve for σ_y.
Now, divide both sides by 1.8 to find σ_y:

\[ \sigma_y = \frac{4.8}{1.8} \]

\[ \sigma_y = \frac{48}{18} = \frac{8}{3} \approx 2.67 \]

Thus, the value of the standard deviation of Y (σ_y) is approximately 2.67.