Moments of Sums and Linear Combinations of Random Variables

Understand Moments of Sums and Linear Combinations of Variables

Key formulas to remember

Mean of a linear combination

E(aX + bY + c) = aE(X) + bE(Y) + c

Always true. No independence needed.

Variance of a weighted sum

Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X, Y)

A constant added to the sum does not change the variance.

Covariance and correlation link

Cov(X, Y) = ρ·σX·σY

Correlation is covariance scaled to lie between −1 and +1.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2

Weights are the portfolio weights. Volatility is σp = √σp².

Variance of a difference

Var(X − Y) = σX² + σY² − 2Cov(X, Y)

The sign of the covariance term flips.

Independent or uncorrelated case

Var(aX + bY) = a²σX² + b²σY²

Valid when Cov(X, Y) = 0. Independence implies zero covariance, but not the reverse.

Matrix form

σp² = wᵀΣw

Σ is the covariance matrix. Diagonal entries are variances.

How to solve Moments of Sums and Linear Combinations of Variables questions

1. 1Write the combination, for example Z = aX + bY, and identify the weights a and b, including any negative sign.
2. 2Compute the mean: aE(X) + bE(Y).
3. 3Check whether the question gives variance, standard deviation, covariance or correlation. Convert everything to variances and covariance.
4. 4If correlation is given, find Cov = ρσXσY.
5. 5Apply Var = a²σX² + b²σY² + 2ab·Cov, keeping the sign of ab.
6. 6Take the square root if the question asks for standard deviation or volatility.
7. 7Sanity check: the result should lie between the values at ρ = −1 and ρ = +1.

Quickest way: Standard deviation shortcut with weighted volatilities

Common mistakes in Moments of Sums and Linear Combinations of Variables

• Adding standard deviations to get portfolio risk.

  Means add, so students assume volatility adds too.

  Fix: Add variances plus the covariance term. Adding σ's works only when ρ = 1 and weights are positive.

• Leaving out the factor 2 on the covariance term.

  Students forget the cross term appears twice in the expansion.

  Fix: Always write 2ab·Cov(X, Y) or 2w1w2ρσ1σ2.

• Forgetting to square the weights or constants.

  Mean scales by a, but variance scales by a².

  Fix: Write a² and b² explicitly. Var(3X) = 9Var(X).

• Using a plus sign for Var(X − Y).

  Students remember the sum formula only.

  Fix: Treat the combination as 1·X + (−1)·Y, so the cross term is 2(1)(−1)Cov = −2Cov.

• Entering volatility where variance is needed, or reporting variance as volatility.

  Questions switch between σ and σ² without warning.

  Fix: Label each input and the final answer, and square root only at the end.

• Assuming zero covariance means independence.

  Both give no covariance term in the variance formula.

  Fix: Zero covariance is enough for the variance formula. Independence is a stronger statement.

Worked examples