12 1. Preparatory material

Now we turn to non-discrete random variables X taking values in some

range R. We say that a random variable is continuous if P(X = x) = 0 for

all x ∈ R (here we assume that all points are measurable). If R is already

equipped with some reference measure dm (e.g., Lebesgue measure in the

case of scalar, vector, or matrix-valued random variables), we say that the

random variable is absolutely continuous if P(X ∈ S) = 0 for all null sets

S in R. By the Radon-Nikodym theorem (see e.g., [Ta2010, §1.10]), we can

thus find a non-negative, absolutely integrable function f ∈

L1(R,

dm) with

R

f dm = 1 such that

(1.5) μX(S) =

S

f dm

for all measurable sets S ⊂ R. More succinctly, one has

(1.6) dμX = f dm.

We call f the probability density function of the probability distribution μX

(and thus, of the random variable X). As usual in measure theory, this

function is only defined up to almost everywhere equivalence, but this will

not cause any diﬃculties.

In the case of real-valued random variables X, the distribution μX can

also be described in terms of the cumulative distribution function

(1.7) FX(x) := P(X ≤ x) = μX((−∞,x]).

Indeed, μX is the Lebesgue-Stieltjes measure of FX , and (in the absolutely

continuous case) the derivative of FX exists and is equal to the probability

density function almost everywhere. We will not use the cumulative distri-

bution function much in this text, although we will be very interested in

bounding tail events such as P(X λ) or P(X λ).

We give some basic examples of absolutely continuous scalar distribu-

tions:

(i) uniform distributions, in which f :=

1

m(I)

1I for some subset I of

the reals or complexes of finite non-zero measure, e.g., an interval

[a, b] in the real line, or a disk in the complex plane.

(ii) The real normal distribution N(μ,

σ2)

= N(μ,

σ2)R

of mean μ ∈

R and variance

σ2

0, given by the density function f(x) :=

1

√

2πσ2

exp(−(x −

μ)2/2σ2)

for x ∈ R. We isolate, in particular,

the standard (real) normal distribution N(0, 1). Random variables

with normal distributions are known as Gaussian random variables.

(iii) The complex normal distribution N(μ,

σ2)C

of mean μ ∈ C and

variance

σ2

0, given by the density function f(z):=

1

πσ2

exp(−|z −

μ|2/σ2).

Again, we isolate the standard complex normal distribu-

tion N(0, 1)C.