Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 896 — #904
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Chapter 20
Deviation from the Mean
Problem 20.26.
The proof of the Pairwise Independent Sampling Theorem 20.4.1 was given for
a sequence R 1 ; R 2 ; : : : of pairwise independent random variables with the same
mean and variance.
The theorem generalizes straighforwardly to sequences of pairwise independent
random variables, possibly with different distributions, as long as all their variances
are bounded by some constant.
Theorem (Generalized Pairwise Independent Sampling). Let X 1 ; X 2 ; : : : be a sequence
of pairwise independent random variables such that VarŒX i b for some
b 0 and all i 1. Let
A n WWD X 1 C X 2 C C X n
;
n
n WWD ExŒA n :
Then for every > 0,
PrŒjA n n j b 2 1 n : (20.28)
(a) Prove the Generalized Pairwise Independent Sampling Theorem.
(b) Conclude that the following holds:
Corollary (Generalized Weak Law of Large Numbers). For every > 0,
Exam Problems
lim
n!1 PrŒjA n n j D 1:
Problem 20.27.
You work for the president and you want to estimate the fraction p of voters in the
entire nation that will prefer him in the upcoming elections. You do this by random
sampling. Specifically, you select a random voter and ask them who they are going
to vote for. You do this n times, with each voter selected with uniform probability
and independently of other selections. Finally, you use the fraction P of voters
who said they will vote for the President as an estimate for p.
(a) Our theorems about sampling and distributions allow us to calculate how confident
we can be that the random variable P takes a value near the constant p. This
calculation uses some facts about voters and the way they are chosen. Indicate the
true facts among the following:
1. Given a particular voter, the probability of that voter preferring the President
is p.