Correlation Formula

D

The correlation between two random variables X and Y is Cov(X,Y)/[(sigma_X)*(sigma_Y)].

The formula for the correlation between two variables X and Y is given by option D: Cov(X, Y) / (sigma_X * sigma_Y).

Let's break down the components of this formula:

1. Cov(X, Y): Covariance is a measure of how two variables vary together. It indicates the direction of the linear relationship between X and Y. If Cov(X, Y) is positive, it suggests a positive linear relationship, meaning that as X increases, Y tends to increase as well. If Cov(X, Y) is negative, it suggests a negative linear relationship, indicating that as X increases, Y tends to decrease. A covariance of zero implies no linear relationship between X and Y.

2. sigma_X: This represents the standard deviation of variable X. The standard deviation is a measure of the dispersion or spread of data points around the mean of X. It gives an indication of the variability of X values.

3. sigma_Y: This represents the standard deviation of variable Y. Similar to sigma_X, it measures the dispersion or spread of data points around the mean of Y.

To calculate the correlation between X and Y, we divide the covariance by the product of the standard deviations (sigma_X and sigma_Y). This normalization ensures that the correlation coefficient ranges between -1 and +1, providing a standardized measure of the linear relationship between X and Y.

Option D, Cov(X, Y) / (sigma_X * sigma_Y), correctly represents the formula for correlation. Options A, B, and C have incorrect formulations of the correlation formula.

• Option A, Cov(X, Y) / [(sigma_X) * (sigma_Y)]^2, incorrectly squares the denominator.
• Option B, [Cov(X, Y)]^0.5, represents the square root of the covariance, which is not the correct calculation for correlation.
• Option C, (sigma_X) * (sigma_Y) / Cov(X, Y), reverses the numerator and denominator, which is incorrect.

Therefore, the correct answer is option D, Cov(X, Y) / (sigma_X * sigma_Y).