Mathematics for Computer Science

19.3. Properties of Variance 797

Here we use the notation Ex^2 ŒRças shorthand for.ExŒRç/^2.

Proof. LetDExŒRç. Then

VarŒRçDExŒ.RExŒRç/^2 ç (Def 19.2.2 of variance)

DExŒ.R/^2 ç (def of)

DExŒR^2 2RC^2 ç

DExŒR^2 ç2ExŒRçC^2 (linearity of expectation)

DExŒR^2 ç2^2 C^2 (def of)

DExŒR^2 ç^2

DExŒR^2 çEx^2 ŒRç: (def of)



A simple and very useful formula for the variance of an indicator variable is an
immediate consequence.

Corollary 19.3.2.IfBis a Bernoulli variable wherepWWDPrŒBD1ç, then

VarŒBçDpp^2 Dp.1p/: (19.6)

Proof. By Lemma 18.4.2, ExŒBçDp. ButBonly takes values 0 and 1, soB^2 DB
and equation (19.6) follows immediately from Lemma 19.3.1. 

19.3.2 Variance of Time to Failure

According to Section 18.4.6, the mean time to failure is1=pfor a process that fails
during any given hour with probabilityp. What about the variance?
By Lemma 19.3.1,
VarŒCçDExŒC^2 ç.1=p/^2 (19.7)

so all we need is a formula for ExŒC^2 ç.
Reasoning aboutCusing conditional expectation worked nicely in Section 18.4.6
to find mean time to failure, and a similar approach works forC^2. Namely, the ex-
pected value ofC^2 is the probability,p, of failure in the first hour times 12 , plus
the probability,.1p/, of non-failure in the first hour times the expected value of