Question

Variance of the sum of two statistically independent random variables \(X\) and \(Y\), \( \sigma^2_{X+Y} \), is

• A. \( \sigma_X^2 + \sigma_Y^2 \).
• B. \( \sigma_X^2 + \sigma_Y^2 + 2\sigma_{XY} \).
• C. \( \sigma_X^2 + \sigma_Y^2 + \sigma_{XY} \).
• D. \( \sigma_X^2 + \sigma_Y^2 - \sigma_{XY} \).

Hint:

When variables are independent, covariance vanishes and variances simply add.

Answer 1

The Correct Option is A

The variance of the sum of two random variables is given by the general formula: \[ \mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\,\mathrm{Cov}(X,Y). \] The covariance \(\mathrm{Cov}(X,Y)\) reflects how the two variables vary together. If they are statistically independent: \[ \mathrm{Cov}(X,Y) = 0. \] This is because independence implies that knowledge of one variable gives no information about the other; hence their joint variability is zero. Substituting this into the general formula: \[ \mathrm{Var}(X + Y) = \sigma_X^2 + \sigma_Y^2. \] This rule is extremely important in engineering, statistics, mining risk assessment, and reliability analysis. For example, when combining measurement errors, sensor noise, or independent geological variables, the variances add directly without any cross-term. Thus, option (A) is correct.