Chapter 3
Random Variables and Univariate Probability
Distributions
Abstract:
In this and next chapters, we will use advanced calculus to formalize the probability
theory introduced in Chapter 2. The use of mathematics enables us to investigate probability
more deeply. A number of quantitative-oriented probability concepts will be introduced. In this
chapter, we ﬁrst introduce the concept of a random variable and characterize the probability
distributions of a random variable and functions of a random variable by the cumulative dis-
tribution function, the probability mass function or probability density function, the moment
generating function and the characteristic function. We also introduce a class of moments and
discuss their relationships with a probability distribution. This chapter focuses on univariate
distributions.
Key words:
Continuous random variable, Cumulative distribution function, Discrete random
variable, Kurtosis, Mean, Moment generating function, Moments, Probability density function,
Probability mass function, Random variable, Skewness, Variance
3.1 Random Variables and Distribution Functions
Recall that the probability space triple (
S,
B
,P
) completely characterizes a random experiment.
In general, the space
S
and the associated
σ
-algebra
B
diﬀer according to the natures of random
experiments. For example, when one throws a coin, the sample space
S
=
{
H,T
}
,
where
H
denotes head, and
T
denotes tail; for the election of some candidate,
S
=
{
Win, Fail
}
; if one
throws three coins, then
S
=
{
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
}
.
It is inconvenient to work with diﬀerent sample spaces. In particular, a sample space
S
may
be tedious to describe if the elements of
S
are not real numbers. In many experiments, it is
easier to deal with a summary variable than with the original probability structure. To develop
a uniﬁed probability theory, we need to unify diﬀerent sample spaces. For this purpose, we need
to formulate a rule, or a set of rules, by which elements of
S
may be represented by numbers.
This can be achieved by assigning a real number to each possible outcome in S. In other words,
we construct a mapping from the original sample space
S
to a new sample space Ω, a set of
real numbers. This transformation is called a random variable. A random variable is a function
deﬁned on a sample space. Its purpose is to facilitate the solution of a problem by transferring
considerations to a new probability space with a simpler or more convenient structure.
On the other hand, in many applications, we are interested only in a particular aspect of the
outcomes of experiments, rather than the outcomes themselves. For example, when we roll a
number of dice, we are usually interested in the total number obtained, and not in the outcome
of each die. In such applications, a suitably deﬁned random variables will better serve for our
purpose.