Examples of Random Variables
1. Let the random variable
X
be the number of heads in
n
coin flips. The sample
space is
Ω
=
{
H, T
}
n
, the possible outcomes of
n
coin flips; then
X
∈
{
0
,
1
,
2
, . . . , n
}
2. Let
Ω
=
R
, the real numbers. Define random variables
X
and
Y
as follows:
a.
X
(
ω
) =
ω
b.
Y
(
ω
) =
+1
ω
≥
0

1
otherwise
3. Packet arrival times in the interval
(0
, T
]
. Here
Ω
is the set of all finite length
strings
(
t
1
, t
2
, . . . , t
n
)
∈
(0
, T
]
*
, where
t
1
≤
t
2
≤
· · ·
≤
t
n
. Define the random
variable
X
to be
n
, the length of the string; then
X
∈
{
0
,
1
,
2
,
3
, . . .
}
4. Let
X
be the service time at a router. If the bu
ff
er is empty the packet is served
immediately, i.e.,
X
= 0
. If it is not empty, the service time
X >
0
is a positive
real number
EE 278: Random Variables
2 – 3
Specifying a Random Variable
•
Specifying a random variable means being able to determine the probability that
X
∈
A
for any Borel set
A
⊂
R
, in particular, for any interval
(
a, b
]
•
To do so, consider the
inverse image
of
A
under
X
, i.e.,
{
ω
:
X
(
ω
)
∈
A
}
R
set
A
inverse image of
A
under
X
(
ω
)
•
Since
X
∈
A
i
ff
ω
∈
{
ω
:
X
(
ω
)
∈
A
}
,
P(
{
X
∈
A
}
) = P(
{
ω
:
X
(
ω
)
∈
A
}
) = P
{
ω
:
X
(
ω
)
∈
A
}
Shorthand:
P(
{
set description
}
) = P
{
set description
}
EE 278: Random Variables
2 – 4