Mathematics for Computer Science

18.4. Properties of Variance 627



It’s even simpler to prove that adding a constant does not change the variance, as
the reader can verify:

Theorem 18.4.5.LetRbe a random variable, andba constant. Then

VarŒRCbçDVarŒRç: (18.10)
Recalling that the standard deviation is the square root of variance, this implies
that the standard deviation ofaRCbis simplyjajtimes the standard deviation of
R:

Corollary 18.4.6.
aRCbDjajR:

18.4.4 Variance of a Sum

In general, the variance of a sum is not equal to the sum of the variances, but
variances do add forindependentvariables. In fact,mutualindependence is not
necessary:pairwiseindependence will do. This is useful to know because there are
some important situations involving variables that are pairwise independent but not
mutually independent.

Theorem 18.4.7.IfR 1 andR 2 are independent random variables, then

VarŒR 1 CR 2 çDVarŒR 1 çCVarŒR 2 ç: (18.11)

Proof. We may assume that ExŒRiçD 0 foriD1;2, since we could always replace
RibyRiExŒRiçin equation (18.11). This substitution preserves the indepen-
dence of the variables, and by Theorem 18.4.5, does not change the variances.
Now by Lemma 18.4.1, VarŒRiçDExŒR^2 içand VarŒR 1 CR 2 çDExŒ.R 1 CR 2 /^2 ç,
so we need only prove

ExŒ.R 1 CR 2 /^2 çDExŒR^21 çCExŒR 22 ç: (18.12)

But (18.12) follows from linearity of expectation and the fact that

ExŒR 1 R 2 çDExŒR 1 çExŒR 2 ç (18.13)

sinceR 1 andR 2 are independent:

ExŒ.R 1 CR 2 /^2 çDExŒR 12 C2R 1 R 2 CR^22 ç

DExŒR 12 çC 2 ExŒR 1 R 2 çCExŒR 22 ç

DExŒR 12 çC 2 ExŒR 1 çExŒR 2 çCExŒR 22 ç (by (18.13))

DExŒR 12 çC 2 
 0 
 0 CExŒR 22 ç

DExŒR 12 çCExŒR^22 ç