Moments for Linear Combinations of Independent Random Variables

An actuary needs $E[S]$ and $\text{Var}(S)$ for $S=c_1{X}_1+\cdots +c_{20}{X}_{20}$ covering 20 independent risks. Building the full distribution is impractical, the moment formulas for linear combinations provide an exact shortcut.

For any random variables $X_1,\dots ,X_n$ (independent or not) and constants $a_1,\dots ,a_n,b$:

$$E\left[\sum _{i=1}^n{a}_i{X}_i+b\right]=\sum _{i=1}^n{a}_{i}E[X_i]+b$$

Linearity of expectation holds always, no independence required. This is one of the most powerful tools in probability.

For any random variables $X_1,\dots ,X_n$:

$$\text{Var}\left(\sum _{i=1}^n{a}_i{X}_i+b\right)=\sum _{i=1}^n{a}_i^2\text{Var}(X_i)+2\sum _{i<j}{a}_i{a}_j\text{Cov}(X_i,X_j)$$

HIGH-FREQUENCY: The constant $b$ does NOT affect variance. The coefficient $a_i$ is SQUARED in the variance terms.

When $X_1,\dots ,X_n$ are independent, all covariance terms vanish:

$$\text{Var}\left(\sum _{i=1}^n{a}_i{X}_i\right)=\sum _{i=1}^n{a}_i^2\text{Var}(X_i)$$

Common mistakes

• Forgetting to square the coefficient in variance. For $2X_1$, the variance contribution is $4\cdot \text{Var}(X_1)=16$, not $2\cdot 4=8$. Trap: 8 appears when you multiply the coefficient by the variance instead of squaring it.
• Including the constant in the variance. $\text{Var}(S+10)=\text{Var}(S)$. Adding 10 shifts the distribution but does not spread it. Including $10^2=100$ in the variance gives $\text{Var}(S)=98+100=198$. Trap: 198.
• Not squaring the negative coefficient for differences. $\text{Var}(-X_2)=(-1{)}^2\text{Var}(X_2)=\text{Var}(X_2)$, not $-\text{Var}(X_2)$. Variance is always non-negative.

Bottom line

• Mean is linear (always): $E[\sum a_i{X}_i+b]=\sum a_{i}E[X_i]+b$, no independence needed
• Variance under independence: $\text{Var}(\sum a_i{X}_i)=\sum a_i^2\text{Var}(X_i)$, coefficients SQUARED
• Constants vanish from variance: $\text{Var}(S+b)=\text{Var}(S)$
• Negative coefficients stay positive: $(-a{)}^2=a^2$, so every variance contribution is non-negative

Exam shortcut

Set up a variance table with columns: Variable, Coefficient, Coeff Squared, Variance, Product. Fill each row, sum the Product column, then add covariance terms if needed. The variance of a sum must be positive, a negative result means a sign error. "Mean is Linear, Variance is Quadratic, Constants Vanish." Mean: coefficients enter linearly, constant enters additively. Variance: coefficients are squared, constant disappears. "The 1/n rule" for sample means: $\text{Var}(\bar{X})={\sigma }^2/n$. $\text{SD}(\bar{X})=\sigma /\sqrt{n}$.

The full lesson (about 2,932 words, 20 min read) adds 3 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 3h

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