The variance of the difference between two independent variables equals the sum of their variances. Which option expresses this correctly?

• A. The variance of the sum equals the sum of the variances.
• B. The variance of the difference equals the product of the variances.
• C. The variance of the difference equals the sum of the variances.
• D. The variance of the sum equals the difference of the variances.

Answer: (C)

Independent variables have variability that adds when you look at their combined outcome. When two variables are independent, the covariance is zero, so the cross-term drops out in the variance calculation. The general rule is Var(X ± Y) = Var(X) + Var(Y) for independent X and Y. Therefore, the variance of the difference is the sum of their variances. For example, if Var(X) = 4 and Var(Y) = 9, then Var(X − Y) = 4 + 9 = 13. The other expressions—such as using a product or mixing plus/minus with a difference of variances—don’t follow this rule.