What is the formula for the variance of a linear combination of random variables?

What is the formula for the variance of a linear combination of random variables?

• A. Var(aX + bY) = a^2Var(X) + b^2Var(Y)
• B. Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)
• C. Var(aX + bY) = Cov(aX,bY)
• D. Var(aX + bY) = 0 if X and Y are independent

Answer: (B)

The formula for the variance of a linear combination of random variables is accurately represented by the expression that incorporates the variances of the individual variables as well as the covariance between them. When combining two random variables \(X\) and \(Y\) scaled by constants \(a\) and \(b\), the variance of the linear combination \(aX + bY\) can be derived from the properties of variance and covariance. Specifically, when both variables are included, the expression accounts for the variability within each variable as captured by their variances \(Var(X)\) and \(Var(Y)\), scaled by the squares of their respective coefficients \(a^2\) and \(b^2\). Additionally, the term \(2abCov(X,Y)\) captures how the two variables interact with each other. The covariance term adjusts the overall variance based on how \(X\) and \(Y\) move together: if they are positively correlated, this term increases the variance; if negatively correlated, it decreases the variance. Thus, this comprehensive formula allows for an accurate calculation of the variance for linear combinations of random variables, making it essential for various applications in probability and statistics.

The formula for the variance of a linear combination of random variables is accurately represented by the expression that incorporates the variances of the individual variables as well as the covariance between them. When combining two random variables (X) and (Y) scaled by constants (a) and (b), the variance of the linear combination (aX + bY) can be derived from the properties of variance and covariance.

Specifically, when both variables are included, the expression accounts for the variability within each variable as captured by their variances (Var(X)) and (Var(Y)), scaled by the squares of their respective coefficients (a^2) and (b^2). Additionally, the term (2abCov(X,Y)) captures how the two variables interact with each other. The covariance term adjusts the overall variance based on how (X) and (Y) move together: if they are positively correlated, this term increases the variance; if negatively correlated, it decreases the variance.

Thus, this comprehensive formula allows for an accurate calculation of the variance for linear combinations of random variables, making it essential for various applications in probability and statistics.