1.1. A review of probability theory 13

Later on, we will encounter several more scalar distributions of relevance

to random matrix theory, such as the semicircular law or Marcenko-Pastur

law. We will also of course encounter many matrix distributions (also known

as matrix ensembles) as well as point processes.

Given an unsigned random variable X (i.e., a random variable taking

values in [0, +∞]), one can define the expectation or mean EX as the un-

signed integral

(1.8) EX :=

∞

0

x dμX(x),

which by the Fubini-Tonelli theorem (see e.g. [Ta2011, §1.7]) can also be

rewritten as

(1.9) EX =

∞

0

P(X ≥ λ) dλ.

The expectation of an unsigned variable lies in also [0, +∞]. If X is a

scalar random variable (which is allowed to take the value ∞) for which

E|X| ∞, we say that X is absolutely integrable, in which case we can

define its expectation as

(1.10) EX :=

R

x dμX(x)

in the real case, or

(1.11) EX :=

C

z dμX(z)

in the complex case. Similarly, for vector-valued random variables (note

that in finite dimensions, all norms are equivalent, so the precise choice of

norm used to define |X| is not relevant here). If X = (X1,...,Xn) is a

vector-valued random variable, then X is absolutely integrable if and only

if the components Xi are all absolutely integrable, in which case one has

EX = (EX1,..., EXn).

Examples 1.1.8. A deterministic scalar random variable c is its own mean.

An indicator function I(E) has mean P(E). An unsigned Bernoulli variable

(as defined previously) has mean p, while a signed or lazy signed Bernoulli

variable has mean 0. A real or complex Gaussian variable with distribution

N(μ,

σ2)

has mean μ. A Poisson random variable has mean λ; a geometric

random variable has mean p. A uniformly distributed variable on an interval

[a, b] ⊂ R has mean

a+b

2

.

A fundamentally important property of expectation is that it is linear: if

X1,...,Xk are absolutely integrable scalar random variables and c1,...,ck

are finite scalars, then c1X1 + · · · + ckXk is also absolutely integrable and

(1.12) Ec1X1 + · · · + ckXk = c1EX1 + · · · + ckEXk.