Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 890 — #898
890
Chapter 20
Deviation from the Mean
Hint: Suppose the sample space outcomes are ! 1 ; ! 2 ; : : : ; ! n , and let
p WWD .p 1 ; p 2 ; : : : ; p n / where p i D p PrŒ! i ;
r WWD .r 1 ; r 2 ; : : : ; r n / where r i D jR.! i / j p PrŒ! i :
As usual, let v w WWD P n
iD1 v iu i denote the dot product of n-vectors v; w, and let
jvj be the norm of v, namely, p v v.
Then verify that
jpj D 1; jrj D ; and ExΠjR j D r p:
Problem 20.17.
Prove the following “one-sided” version of the Chebyshev bound for deviation
above the mean:
Lemma (One-sided Chebyshev bound).
PrŒR
ExŒR x
VarŒR
x 2 C VarŒR :
Hint: Let S a WWD .R ExŒR C a/ 2 , for 0 a 2 R. So R ExŒR x
implies S a .x C a/ 2 . Apply Markov’s bound to PrŒS a .x C a/ 2 . Choose a to
minimize this last bound.
Problem 20.18.
Prove the pairwise independent additivity of variance Theorem 20.3.8: If R 1 ; R 2 ; : : : ; R n
are pairwise independent random variables, then
VarŒR 1 C R 2 C C R n D VarŒR 1 C VarŒR 2 C C VarŒR n : (*)
Hint: Why is it OK to assume ExŒR i D 0?
Exam Problems
Problem 20.19.
You are playing a game where you get n turns. Each of your turns involves flipping
a coin a number of times. On the first turn, you have 1 flip, on the second turn you
have two flips, and so on until your nth turn when you flip the coin n times. All the
flips are mutually independent.
The coin you are using is biased to flip Heads with probability p. You win a turn
if you flip all Heads. Let W be the number of winning turns.