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-
(1.8) EX :=
which by the Fubini-Tonelli theorem (see e.g. [Ta2011, §1.7]) can also be
(1.9) EX =
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 :=
in the real case, or
(1.11) EX :=
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
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 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.