In assessing variance, how is var(aX + bY) evaluated?

In assessing variance, how is var(aX + bY) evaluated?

• A. It considers the variances of X and Y alone
• B. It includes the covariance term 2Cov(X,Y)
• C. It is equal to a^2Var(X) + b^2Var(Y)
• D. It is independent of the coefficients a and b

Answer: (C)

When evaluating the variance of a linear combination of random variables, specifically the expression var(aX + bY), the correct approach involves utilizing the properties of variance and covariance. The formula used to calculate the variance of a linear combination of two variables X and Y is given by var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y). Therefore, the variance is influenced not just by the variances of the individual variables (X and Y), but also by their covariance, which accounts for how the two variables move together. However, the choice stating that it is equal to a^2Var(X) + b^2Var(Y) does not include the covariance term. This means that while part of combining the variances is accurately depicted, the full evaluation of var(aX + bY) requires consideration of the covariance. Thus, the conclusion that it omits the covariance contribution leads to an incomplete assessment. Overall, while the response indicated is partially correct regarding the use of variances, it misses a crucial dimension of the broader formula involving covariance. Thus, understanding variance in this context requires recognizing all contributing components, which shapes a more comprehensive insight into how variances

When evaluating the variance of a linear combination of random variables, specifically the expression var(aX + bY), the correct approach involves utilizing the properties of variance and covariance.

The formula used to calculate the variance of a linear combination of two variables X and Y is given by var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y). Therefore, the variance is influenced not just by the variances of the individual variables (X and Y), but also by their covariance, which accounts for how the two variables move together.

However, the choice stating that it is equal to a^2Var(X) + b^2Var(Y) does not include the covariance term. This means that while part of combining the variances is accurately depicted, the full evaluation of var(aX + bY) requires consideration of the covariance. Thus, the conclusion that it omits the covariance contribution leads to an incomplete assessment.

Overall, while the response indicated is partially correct regarding the use of variances, it misses a crucial dimension of the broader formula involving covariance. Thus, understanding variance in this context requires recognizing all contributing components, which shapes a more comprehensive insight into how variances