14 1. Preparatory material

By the Fubini-Tonelli theorem, the same result also applies to infinite sums

∑∞

i=1

ciXi provided that

∑∞

i=1

|ci|E|Xi| is finite.

We will use linearity of expectation so frequently in the sequel that we

will often omit an explicit reference to it when it is being used. It is im-

portant to note that linearity of expectation requires no assumptions of

independence or

dependence4

amongst the individual random variables Xi;

this is what makes this property of expectation so powerful.

In the unsigned (or real absolutely integrable) case, expectation is also

monotone: if X ≤ Y is true for some unsigned or real absolutely integrable

X, Y , then EX ≤ EY . Again, we will usually use this basic property without

explicit mentioning it in the sequel.

For an unsigned random variable, we have the obvious but very useful

Markov inequality

(1.13) P(X ≥ λ) ≤

1

λ

EX

for any λ 0, as can be seen by taking expectations of the inequality

λI(X ≥ λ) ≤ X. For signed random variables, Markov’s inequality becomes

(1.14) P(|X| ≥ λ) ≤

1

λ

E|X|.

Another fact related to Markov’s inequality is that if X is an unsigned

or real absolutely integrable random variable, then X ≥ EX must hold

with positive probability, and also X ≤ EX must also hold with positive

probability. Use of these facts or (1.13), (1.14), combined with monotonicity

and linearity of expectation, is collectively referred to as the first moment

method. This method tends to be particularly easy to use (as one does not

need to understand dependence or independence), but by the same token

often gives sub-optimal results (as one is not exploiting any independence

in the system).

Exercise 1.1.1 (Borel-Cantelli lemma). Let E1,E2,... be a sequence of

events such that

∑

i

P(Ei) ∞. Show that almost surely, at most finitely

many of the events Ei occur at once. State and prove a result to the effect

that the condition

∑

i

P(Ei) ∞ cannot be weakened.

If X is an absolutely integrable or unsigned scalar random variable, and

F is a measurable function from the scalars to the unsigned extended reals

[0, +∞], then one has the change of variables formula

(1.15) EF (X) =

R

F (x) dμX(x)

4We

will define these terms in Section 1.1.3.