Mathematics for Computer Science

18.4. Properties of Variance 625

Proof. LetDExŒRç. Then

VarŒRçDExŒ.RExŒRç/^2 ç (Def 18.3.2 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)



For example, ifBis a Bernoulli variable wherepWWDPrŒBD1ç, then

Lemma 18.4.2.
VarŒBçDpp^2 Dp.1p/: (18.6)

Proof. By Lemma 17.4.2, ExŒBçDp. But sinceBonly takes values 0 and 1,
B^2 DB. So Lemma 18.4.2 follows immediately from Lemma 18.4.1. 

18.4.2 Variance of Time to Failure

According to section 17.4.6, the mean time to failure is1=pfor a process that fails
during any given hour with probabilityp. What about the variance?
By Lemma 18.4.1,
VarŒCçDExŒC^2 ç.1=p/^2 (18.7)

so all we need is a formula for ExŒC^2 ç.
Reasoning aboutCusing conditional expectation worked nicely in section 17.4.6
to find mean time to failure,, and a similar approach worksC^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