To find the variance of Y , given the coefficient of correlation, the covariance, and the variance of X , we can use the relationship between these quantities.
Given data:
• Coefficient of correlation ( r ) between X and Y : r = 0.4
• Covariance ( \text{Cov}(X, Y) ) between X and Y : \text{Cov}(X, Y) = 10
• Variance of X ( \sigma_X^2 ): \sigma_X^2 = 9
The formula relating these quantities is:
r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}
Where \sigma_X and \sigma_Y are the standard deviations of X and Y , respectively. We need to find the variance of Y ( \sigma_Y^2 ).
First, compute \sigma_X from the given variance of X :
\sigma_X = \sqrt{\sigma_X^2} = \sqrt{9} = 3
Substitute the known values into the correlation formula and solve for \sigma_Y :
0.4 = \frac{10}{3 \sigma_Y}
Solve for \sigma_Y :
0.4 \sigma_Y = \frac{10}{3}
\sigma_Y = \frac{10}{3 \times 0.4}
\sigma_Y = \frac{10}{1.2}
\sigma_Y = \frac{10 \times 10}{12} = \frac{100}{12} = \frac{25}{3} \approx 8.33
Now, square \sigma_Y to find the variance of Y :
\sigma_Y^2 = \left(\frac{25}{3}\right)^2 = \frac{625}{9} \approx 69.44
Thus, the variance of Y is:
\sigma_Y^2 = \frac{625}{9} \approx 69.44
This is the variance of Y based on the given information.
