ECE 6010: Lecture 2 – More on Random Variables
5
where
R
ab
is the semi-infinite rectangle
R
a,b
=
{
(
x, y
)
∈
R
2
:
x
≤
a, y
≤
b
}
.
2
Properties of the joint c.d.f.:
1.
lim
a,b
→∞
F
X,Y
(
a, b
) = 1
.
2.
lim
a
→-∞
F
X,Y
(
a, b
) = 0 = lim
b
→-∞
F
X,Y
(
a, b
)
.
3.
lim
a
→∞
F
X,Y
(
a, b
) =
F
Y
(
b
)
, the marginal c.d.f. of
Y
.
lim
b
→∞
F
X,Y
(
a, b
) =
F
X
(
a
)
, the marginal c.d.f. of
X
.
4.
F
XY
(
a, b
)
is continuous “from the northeast.”
5.
F
XY
(
x, y
)
is montonically increasing (or, more precisely, nondecreasing) in both
variables.
Any function with these properties is a legitimate c.d.f., and completely characterizes the
family of joint c.d.f.s.
Joint discrete r.v.s
Definition 4
If
X, Y
are discrete r.v.s taking values in sets
{
x
1
, . . .
}
and
{
y
1
, . . . ,
}
, re-
spectively, then
(
X, Y
)
forms a discrete bivariate r.v. and its joint p.m.f. is defined by
p
XY
(
a, b
) =
P
(
X
=
a, Y
=
b
)
2
Properties of
p
XY
:
1.
p
XY
≥
0
, and
p
XY
(
a, b
) = 0
if
a
6∈ {
x
1
, . . .
}
or
b
6∈ {
y
1
, . . .
}
2.
∑
∞
i
=1
∑
∞
j
=1
p
XY
(
x
i
, y
j
) = 1
.
3.
F
XY
(
a, b
) =
∑
{
x
i
,y
i
}
:
x
i
≤
a,y
j
≤
b
}
p
XY
(
x
i
, y
j
)
4. Marginals:
P
X
(
x
i
) =
X
j
p
XY
(
x
i
, y
j
)
P
Y
(
y
j
) =
X
i
p
XY
(
x
i
, y
j
)
Joint continuous r.v.s
Definition 5
X
and
Y
are
jointly continuous
r.v.s if there is a function
f
XY
:
R
2
→
R
2
such that
F
XY
(
a, b
) =
Z
b
∞
Z
a
∞
f
XY
(
x, y
)
dxdy
for all
(
a, b
)
∈
R
2
. The function
f
XY
is called the
joint p.d.f.
of
X
and
Y
(when it exists).
2
Properties of joint p.d.f.:
1.
f
XY
≥
0
.
2.
R
∞
-∞
R
∞
-∞
f
XY
(
x, y
)
dx, dy
= 1
.