THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 PROBABILITY AND STATISTICS I, FALL 2010
EXAMPLE CLASS 3
Random Variable
Elements of Theory
A
function
(with some requirement)
defined on the sample space
{ }
is called a
random variable.

Domain is the sample space

Range is
usually
a numbers set, e.g.,
or its subsets, for easy manipulation.

The range is called the
state space
of the random variable. There is no intrinsic difference on the nature
between a sample space and a state space
—
they are just two sets with some requirement, called
“measurability.” They are just domain and range of a “function” with some requireme
nt, called
“measurability.”
The variable perspective
is adopted by an observer of a random experiment. The observer is only able to
observe/know/measure/obtain information based on the state space. For the observer, all she could see is a variable
dancing (randomly) on the state space. This is the perspective that we will primarily study in this course.
The function perspective is adopted by someone who would have a “divine” capacity in understanding (a
deterministic part of) the design of the random mechanism, in particular, her capacity in seeing the existence of an
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
underlying sample space as the domain of a function sending elements of the domain to the state space. You will be
studying this perspective in an advanced course of probability.
There is a law governing how any random variable to be observed in its state space. The law is a probabilistic one,
called the probability distribution of a random variable. There are two qualifications for any realvalued function
to be a probability density/mass function, aka, probability function:
1)
for any
,
2)
∫
or
∑
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 SMSLee
 Statistics, Probability, Probability distribution, Probability theory, probability density function, Cumulative distribution function, University of Hong Kong

Click to edit the document details