Moments for Linear Combinations

Study guide: Exam P-1

Moments for Linear Combinations give the expected value and variance of any weighted sum W = a_1 X_1 + a_2 X_2 + \cdots + a_n X_n of random variables. These results underlie the Central Limit Theorem and are used to compute the Expected Value and Variance of aggregate loss models.

E[W] = a_1 E[X_1] + a_2 E[X_2] + \cdots + a_n E[X_n]

\text{Var}(W) = \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)

• When the variables are Independent Random Variables, all covariance terms vanish:

\text{Var}(W) = \sum_{i=1}^n a_i^2 \,\text{Var}(X_i)

Portfolio of Two Risks X_1 \sim (E=100, \text{Var}=400) and X_2 \sim (E=150, \text{Var}=900), independent. Find E[X_1+X_2] and \text{Var}(X_1+X_2).

Answer E[X_1+X_2] = 100 + 150 = 250 \text{Var}(X_1+X_2) = 400 + 900 = 1300 \implies \text{SD} = \sqrt{1300} \approx 36.06