Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 877 — #885
20.6. Sums of Random Variables 877
Algebra aside, there is a brilliant idea in this proof: in this context, exponentiating
somehow supercharges the Markov bound. This is not true in general! One
unfortunate side-effect of this supercharging is that we have to bound some nasty
expectations involving exponentials in order to complete the proof. This is done in
the two lemmas below, where variables take on values as in Theorem 20.6.1.
Lemma 20.6.2.
h i
Ex c T e .c 1/ ExŒT :
Proof.
h i i
Ex c T D Ex
hc T 1CCT n
(def of T )
h i
D Ex c T 1
c T n
h i
D Ex c T 1
ExŒc T n
(independent product Cor 19.5.7)
e .c 1/ ExŒT 1 e .c 1/ ExŒT n
D e .c 1/.ExŒT 1CCExŒT n /
D e .c
1/ ExŒT 1CCT n
D e .c 1/ ExŒT :
(Lemma 20.6.3 below)
(linearity of ExŒ)
The third equality depends on the fact that functions of independent variables are
also independent (see Lemma 19.2.2).
Lemma 20.6.3.
ExŒc T i
e .c
1/ ExŒT i
Proof. All summations below range over values v taken by the random variable T i ,
which are all required to be in the interval Œ0; 1.
ExŒc T i
D X c v PrŒT i D v
(def of ExŒ)
X .1 C .c 1/v/ PrŒT i D v (convexity—see below)
D X PrŒT i D v C .c
D X PrŒT i D v C .c
D 1 C .c 1/ ExŒT i
e .c
1/ ExŒT i
1/v PrŒT i D v
1/ X v PrŒT i D v
(since 1 C z e z ):