Let \(X\) and \(Y\) be two random variables with mean 0, variance 1, and correlation coefficient \(\frac{1}{3}\). Then the value of \(\text{Var}(X + 3Y)\) is equal to
GATE XE - 2025,
Answer: D
The variance is 12.
\[ \begin{aligned} \text{Var}(X+3Y) &= \text{Var}(X) + 9\,\text{Var}(Y) + 6\,\text{Cov}(X,Y) \\ &= 1 + 9 + 6 \times \tfrac{1}{3} = 12 \end{aligned} \]
With unit variances, the covariance equals the correlation coefficient, 1/3.
10 is the answer for uncorrelated variables, and 11 counts the covariance only once (3 × 1/3) instead of twice. 9 ignores Var(X) as well.
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