How to Solve Multivariate Probability — Multivariate — covariance of linear combinations Questions on Exam P

Variance responds to squared linear coefficients, while an added constant contributes no variability. Independence removes covariance terms, so the displayed marginal variances completely determine each requested linear-combination summary. For independent random variables \(X\) and \(Y\) and constants \(b, d, k\), the linear combination variance is \(\operatorname{Var}(bX+dY+k)=b^2\operatorname{Var}(X)+d^2\operatorname{Var}(Y)\), because constants contribute zero variance and covariance is zero. Applying the rule with \(b=2\), \(d=-1\), and \(k=-5\): \(\operatorname{Var}(2X-Y-5)=(2)^2(1)+(-1)^2(2)=6\). Thus \(\operatorname{Var}(2X-Y-5)=6\), which matches the correct option.