## Definition A ==Linear Combination== of random variables $X_1, X_2, \ldots, X_n$ with constants $a_1, a_2, \ldots, a_n$ is $Y = \sum_{i=1}^n a_i X_i$. Key properties: $ E[Y] = \sum_{i=1}^n a_i E[X_i] $ $ \text{Var}(Y) = \sum_{i=1}^n a_i^2 \text{Var}(X_i) + 2\sum_{i < j} a_i a_j \text{Cov}(X_i, X_j) $ If the $X_i$ are independent, the cross-covariance terms vanish. > [!example]- $X$ and $Y$ are independent with $E[X] = 10$, $\text{Var}(X) = 4$, $E[Y] = 6$, $\text{Var}(Y) = 9$. Find $E[3X - 2Y]$ and $\text{Var}(3X - 2Y)$. > $ E[3X - 2Y] = 3(10) - 2(6) = 30 - 12 = 18 $ > Since $X$ and $Y$ are independent: > $ \text{Var}(3X - 2Y) = 9 \cdot \text{Var}(X) + 4 \cdot \text{Var}(Y) = 9(4) + 4(9) = 36 + 36 = 72 $