Digital Signal Processing Reference
In-Depth Information
P
X
(
x
)
p
X
(
x
)
1
1
b
−
a
0
a
b
x
0
a
b
x
(a)
(b)
FIGURE 2.3
Example of a uniform distribution in the interval
(
a
,
b
]
: (a) probability density function of
X
and
(b) cumulative distribution function of
X
.
In this case, the cumulative distribution function is given by
⎨
⎩
0,
x
a
x
−
a
P
X
(
x
)
=
a
,
a
<
x
b
(2.42)
b
−
0,
x
>
b
Figure 2.3 shows the pdf and cdf of a uniformly distributed r.v.
X
.
2.3.2.1 Joint and Conditional Densities
When we work with random models in practical applications, the number
of r.v.'s required to describe the behavior of the events is often greater than
one. In this section, we extend the probabilistic concepts exposed so far to
the case of multiple r.v.'s. Actually, we shall consider in detail exclusively
the particular case of two variables, since the extension to the generic multi-
dimensional case is somewhat direct. If we consider two r.v.'s
X
and
Y
,we
can define the following distribution.
DEFINITION 2.3
The joint distribution function
P
X
,
Y
(
is the probability
that the r.v.
X
is less than or equal to a specified value
x
and that the r.v.
Y
is
less than or equal to a specified value
y
.
x
,
y
)
Mathematically, for
(
−∞
<
X
≤
x
,
−∞
<
Y
≤
y
)
, we write
Pr
X
y
P
X
,
Y
(
x
,
y
)
=
≤
x
,
Y
≤
(2.43)
Notice that (2.43) states that the outcome associated to the joint event is a
point of the
xy
-plane. It is worth mentioning that
X
and
Y
may be considered
as two separate one-dimensional r.v.'s as well as two components of a single
two-dimensional r.v.
