Session 5.6 – Linear Functions of Random Variables

Idea: A linear function looks like L = aX + bY + c (or more variables). Rules you must know: E[L] = aE[X] + bE[Y] + c (linearity of expectation — always true) Var(L) = a²Var(X) + b²Var(Y) + 2ab·Cov(X,Y) (needs Cov/ρ; if independent, Cov=0)

A) Two variables: L = aX + bY + c

μ_X σ_X μ_Y σ_Y ρ assume independent (ρ=0)

a b c simulate N

E[L] = a μ_X + b μ_Y + c = —

Var(L) = a²σ_X² + b²σ_Y² + 2ab·ρ·σ_Xσ_Y = — ; SD(L) = —

If independent: set ρ=0 → Var(L) = a²σ_X² + b²σ_Y².

simulation: mean(L) ≈ — , sd(L) ≈ — (should be close to the formulas above)

B) Three variables (grade): L = w₁·HW + w₂·Mid + w₃·Final

Label 1 μ₁ σ₁ w₁

Label 2 μ₂ σ₂ w₂

Label 3 μ₃ σ₃ w₃

ρ₁₂ ρ₁₃ ρ₂₃ assume independent (all ρ=0)

simulate N

E[L] = Σ wᵢ μᵢ = —

Var(L) = Σ wᵢ² σᵢ² + 2Σ_i<j wᵢ wⱼ ρᵢⱼ σᵢ σⱼ = — ; SD(L) = —

Pair	Term
Total	—

simulation: mean(L) ≈ — , sd(L) ≈ —

Matrix view (Var(L) = wᵀ Σ w)

Σ is the covariance matrix; we compute u = Σw, then wᵀu = Var(L).

Σ (covariance)

w (weights)

Σ w

wᵀ Σ w = — (should equal Var(L) above)

Quick facts

Linearity of expectation (always): E[aX+bY+c] = aE[X]+bE[Y]+c.
Variance needs dependence info: Var(aX+bY) = a²Var(X)+b²Var(Y)+2abCov(X,Y).
Independence ⇒ Cov=0. But Cov=0 does NOT always mean independent (except for bivariate normal).
More variables: Var(Σ cᵢXᵢ) = Σ cᵢ²Var(Xᵢ) + 2Σ_i<j cᵢcⱼCov(Xᵢ,Xⱼ).
Normal closure: Any linear combo of jointly normal variables is normal.

Excel steps (copy/paste)

2 variables (with correlation ρ):

E[L] = a*μx + b*μy + c
Var(L) = a^2*σx^2 + b^2*σy^2 + 2*a*b*ρ*σx*σy
SD(L) = SQRT(Var(L))
Independent case: set ρ=0 → Var(L) = a^2*σx^2 + b^2*σy^2.
3 variables (pairwise correlations ρ12, ρ13, ρ23):

E[L] = SUMPRODUCT(w_range, mu_range)
Var(L) = SUMPRODUCT(w_range^2, var_range) + 2*(w1*w2*ρ12*σ1*σ2 + w1*w3*ρ13*σ1*σ3 + w2*w3*ρ23*σ2*σ3)
Matrix method (general k): Put weights column w and covariance matrix Σ (σᵢσⱼρᵢⱼ; diagonals σᵢ²). Then

E[L] = SUMPRODUCT(w, mu)
Var(L) = MMULT( TRANSPOSE(w), MMULT( Σ, w ) )
(Use dynamic arrays or confirm with Ctrl+Shift+Enter in older Excel.)
Simulate normals (for checking):
Two vars: Z1,Z2 ~ N(0,1). Set X = μx + σx*Z1; Y = μy + σy*(ρ*Z1 + SQRT(1-ρ^2)*Z2). Compute L and use AVERAGE, STDEV.S.