Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 868 — #876
868
Chapter 20
Deviation from the Mean
20.4.2 Pairwise Independent Sampling
The reasoning we used above to analyze voter polling and matching birthdays is
very similar. We summarize it in slightly more general form with a basic result
called the Pairwise Independent Sampling Theorem. In particular, we do not need
to restrict ourselves to sums of zero-one valued variables, or to variables with the
same distribution. For simplicity, we state the Theorem for pairwise independent
variables with possibly different distributions but with the same mean and variance.
Theorem 20.4.1 (Pairwise Independent Sampling). Let G 1 ; : : : ; G n be pairwise
independent variables with the same mean and deviation . Define
Then
S n
Pr
ˇˇˇˇ n
S n WWD
nX
G i : (20.22)
iD1
ˇ x 1
2
:
n x
Proof. We observe first that the expectation of S n =n is :
P n Sn
iD1
Ex D Ex
G
i
(def of S n )
n
n
P n
iD1
D
ExŒG i
(linearity of expectation)
P
n
n
iD1
D
n
D n n D :
The second important property of S n =n is that its variance is the variance of G i
divided by n:
Sn 1
2
Var D VarŒS n (Square Multiple Rule for Variance (20.9))
n n
" nX
#
D 1 n 2 Var G i (def of S n )
iD1
D 1 n 2
nX
VarŒG i
iD1
D 1 n 2 n 2 D 2
(pairwise independent additivity)
n : (20.23)