Chapter 2: Discrete Random Variables
2.1 Definitions
Recall that an experiment consists of a procedure and observations. In this chapter, we
begin to examine probability models that assign numbers to outcomes in the sample space
S
. When we observe one of these numbers, we refer to the observation as a
random variable
.
By convention, we use a capital letter to denote a random variable (for example,
X
or
Y
),
and denote the
range
of the random variable (i.e., the set of possible values of the random
variable) by the letter
S
with a subscript that is the name of the random variable—for
example
S
X
.
Definition 2.1
A
random variable
consists of an experiment with a probability measure
P
[
·
]
defined on a sample space
S
and a function that assigns a real number to each outcome in
the sample space of the experiment.
A random variable can be related to an experiment in different ways.
1.
The random variable can be the observation
. For example, suppose the experiment is
to attach a photo detector to an optical fiber and count the number of photons arriving
in a one nanosecond time interval.
Each observation is a random variable
X
whose
range is
S
X
=
{
0
,
1
,
2
, . . .
}
. In this situation,
S
X
, the range of
X
, and the sample space
S
are identical.
2.
The random variable is a function of the observation
. For example, suppose the ex
periment consists of testing 6 integrated circuits and after each test observing whether
the circuit is accepted (
a
) or rejected (
r
). Each observation is a sequence of 6 letters
where each letter is either
a
or
r
. The sample space consists of
m
n
or 2
6
= 64 possible
sequences. Define a random variable related to this experiment as
N
, the number of
accepted circuits. Assume for outcome 8 (i.e., the 8
th
observation),
s
8
= aaraaa, so
that
N
= 5. The range of
N
is
S
N
=
{
0
,
1
, . . . ,
6
}
.
3.
The random variable is a function of another random variable
. For the example above,
suppose the net revenue
R
obtained for a batch of 6 integrated circuits is $5 for each
circuit accepted minus $7 for each circuit rejected.
When
N
circuits are accepted,
6

N
circuits are rejected so that the net revenue
R
is related to
N
by the function
R
=
g
(
N
) = 5
N

7(6

N
) = 12
N

42 dollars
.
Because
S
N
=
{
0
, . . . ,
6
}
, the range of
R
is
S
R
=
{
42
,

30
,

18
,

6
,
6
,
18
,
30
}
.
Example 1
A stockroom clerk returns three safety helmets at random to three steel mill employees who
had previously checked them. Suppose Smith, Jones, and Brown, in that order, receive one
of the three hats, and define a random variable related to this experiment as
N
, the number
18
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of correct matches.
What is
S
, the sample space for this experiment (i.e., what are the
possible outcomes of this experiment) and what is
S
N
, the range of
N
?
Definition 2.2
Discrete Random Variable
:
X
is a discrete random variable if the range of
X
is a countable set
S
X
=
{
x
1
, x
2
, . . .
}
.
The defining characteristic of a discrete random
variable is that the set of possible values can, in principle, be listed, even though the list
may be infinitely long. Often, but not always, a discrete random variable takes on integer
values.
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 Spring '09
 Probability theory

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