LESSON 5.1 · ABOUT 20 MINUTES
Sums & linear combinations
Combine independent risks without confusing variance and standard deviation.
By the end of this lesson
- Compute moments of a weighted sum.
- Find a discrete sum distribution and combine independent normals.
Moments
E[aX+bY+c]=aE[X]+bE[Y]+c. For independent variables, Var(aX+bY+c)=a²Var(X)+b²Var(Y). Without independence, add 2abCov(X,Y).
Distribution of a discrete sum
For independent discrete X,Y, P(X+Y=s)=Σ_x p_X(x)p_Y(s−x). Enumerate pairs, multiply their probabilities, then add pairs with the same sum.
Normal sums
A linear combination of independent normal variables is normal. Compute its mean and variance first, take the square root for its standard deviation, then standardize. A negative coefficient still contributes a positive squared coefficient to variance.
Var(Σ aᵢXᵢ) = Σ aᵢ²Var(Xᵢ), for independent Xᵢ
Worked example
Independent X,Y have means 2,3 and variances 4,9. Find moments of Z=2X−Y+5.
E[Z]=4−3+5=6. Var(Z)=4×4+9=25, so SD(Z)=5. If X,Y are normal, Z is N(6,25), using variance as the second parameter.