4

Probability Distributions

The deﬁnition of probability distributions

f

X

(

x

) was left aside in

chapter 3. This chapter presents the formulation and properties for

the probability distributions employed in this book: the Normal

distribution for

x 2 R

, the log-normal for

x 2 R

+

, and the Beta for

x 2 (0, 1).

4.1 Normal Distribution

The most widely employed probability distribution is the Normal,

also known as the Gaussian, distribution. In this book, the names

Gaussian and Normal are emp loyed interchangeably when describ-

ing a probability distribu t ion . This section covers the math em at ic al

foundation for the univariate and multivariate Normal and then

details the properties explaining its widespread usage.

4.1.1 Univariate Normal

Univariate Normal

x 2 R : X ⇠N(x; µ,

2

)

4 2 0 2 4

0

0.1

0.2

0.3

µ

µ +

µ

x

f

X

(x)

(a) Probability density function (PDF)

4 2 0 2 4

0

0.2

0.4

0.6

0.8

1

µ

µ +

µ

x

F

X

(x)

(b) Cumulative distribution function (CDF)

Figure 4.1: Representation of the univari-

ate Normal for µ =0, =1.

The probability density function (PDF) for a Normal random

variable is deﬁned over the real numbers

x 2 R

.

X ⇠N

(

x

;

µ,

2

)is

parameterized by its me