$E\!\left[\sum_i c_i X_i\right] = \sum_i c_i\,E[X_i], \qquad \text{Var}\!\left(\sum_i c_i X_i\right) = \sum_i c_i^2\,\text{Var}(X_i)$ $\text{(variance formula holds when the } X_i \text{ are independent)}$ The Expected Value of a linear combination is the linear combination of the individual means, regardless of dependence. The Variance formula above holds only for independent variables; with dependence, cross terms $2c_i c_j \text{Cov}(X_i, X_j)$ must be added for every pair $i \neq j$. > [!example]- Mean and Variance of a Weighted Sum of Claims {💡 Example} > Independent claim amounts $X_1, X_2, X_3$ each have mean 50 and variance 400. Let $L = 2X_1 + X_2 - X_3$. Find $E[L]$ and $\text{Var}(L)$. > > > [!answer]- Answer > > $E[L] = 2(50)+1(50)+(-1)(50) = 100$ > > Since the $X_i$ are independent: > > $\text{Var}(L) = 2^2(400)+1^2(400)+(-1)^2(400) = 1600+400+400 = 2400$