Springer Texts in Business and Economics

A Review of Some Basic Statistical Concepts

2.1 Variance and Covariance of Linear Combinations of Random Variables.

a. Let Y = a + bX, then E(Y) = E(a + bX) = a + bE(X). Hence,

var(Y) = E[Y - E(Y)]2 = E[a + bX - a - bE(X)]2 = E[b(X - E(X))]2 = b2E[X - E(X)]2 = b2 var(X).

Only the multiplicative constant b matters for the variance, not the additive constant a.

b. Let Z = a + bX + cY, then E(Z) = a + bE(X) + cE(Y) and

var(Z) = E[Z - E(Z)]2 = E[a + bX + cY - a - bE(X) - cE(Y)]2 = E[b(X - E(X)) + c(Y - E(Y))]2

= b2E[X-E(X)]2 + c2E[Y-E(Y)]2+2bc E[X-E(X)][Y-E(Y)]

= b2var(X) + c2var(Y) + 2bc cov(X, Y).

c. LetZ = a+bX+cY, andW = d+eX+fY, thenE(Z) = a+bE(X) + cE(Y) E(W) = d + eE(X) + fE(Y)

and

cov(Z, W) = E[Z - E(Z)][W - E(W)]

= E[b(X-E(X))+c(Y-E(Y))][e(X-E(X))+f(Y-E(Y))]

= be var(X) + cf var(Y) + (bf + ce) cov(X, Y).