Databases Reference
In-Depth Information
example,
P
is the probability of seeing a commercial when the television is
turned on (generally, we drop the argument and simply write this as
P
(
X
(ω)
1
)
(
X
1
)
). Similarly,
P
(
programsthatmaycontainne
w
s
)
can be written as
P
(
1
<
X
4
)
, which is substantially
less cumbersome.
A.3 Distribution Functions
Defining the random variable in the way that we did allows us to define a special probability
P
(
X
x
)
. This probability is called the
cumulative distribution function (cdf)
and is denoted
by
F
X
(
, where the random variable is the subscript and the realization is the argument. One
of the primary uses of probability is the modeling of physical processes, and we will find the
cumulative distribution function very useful when we try to describe or model different random
processes. We will see more on this later.
For now, let us look at some of the properties of the
cdf
:
x
)
Property 1:
0
1. This follows from the definition of the
cdf
.
Property 2:
The
cdf
is a monotonically nondecreasing function. That is,
F
X
(
x
)
x
1
x
2
⇒
F
X
(
x
1
)
F
X
(
x
2
)
To show this simply write the
cdf
as the sum of two probabilities:
F
X
(
x
1
)
=
(
x
1
)
=
(
x
2
)
+
(
x
2
<
x
1
)
P
X
P
X
P
X
=
F
X
(
x
2
)
+
P
(
x
1
<
X
x
2
)
F
X
(
x
2
)
Property 3:
lim
n
F
X
(
x
)
=
1
→∞
Property 4:
lim
F
X
(
x
)
=
0
n
→−∞
If we define
Property 5:
x
−
)
=
F
X
(
P
(
X
<
x
)
then
x
−
)
P
(
X
=
x
)
=
F
X
(
x
)
−
F
X
(
