Lecture 7
ORIE3500/5500 Summer2009 Chen
Continuous random vectors
1 Continuous random vectors
The deﬁnition of a continuous random vector is the analog of the deﬁnition of
continuous random variable in the higher dimension. (
X,Y
) is a continuous
random vector if it has a joint density, that is, there exists a bivariate function
f
X,Y
(
x,y
) which satisﬁes
P
[(
X,Y
)
∈
A
] =
Z Z
A
f
X,Y
(
x,y
)
dxdy,
∀
A
⊂
R
2
.
We can draw similarities immediately. The joint cdf and the joint pdf are
related as
F
X,Y
(
x,y
) =
Z
x
∞
Z
y
∞
f
X,Y
(
s,t
)
dsdt
and
∂
2
∂x∂y
F
X,Y
(
x,y
) =
f
X,Y
(
x,y
)
.
If
B
= (
b
1
,b
2
] and
C
= (
c
1
,c
2
] are one dimensional sets and
A
is their
cartesian product(the rectangle formed with sides
B
and
C
), then
P
[(
X,Y
)
∈
A
] =
P
[
X
∈
B,Y
∈
C
] =
Z
b
2
b
1
Z
c
2
c
1
f
X,Y
(
x,y
)
dxdy.
Recall that for a continuous random variable
X
, we have
∂
∂x
F
X
(
x
) =
f
X
(
x
)
. Then
P
(
x
≤
X
≤
x
+
δ
) =
F
X
(
x
+
δ
)

F
X
(
x
)
≈
f
X
(
x
)
δ,
which means
f
X
(
x
)is the probability per unit length. Now for two dimen
sional random vectors,
P
(
x
≤
X
≤
x
+
δ,y
≤
Y
≤
y
+
δ
) =
Z
y
+
δ
y
Z
x
+
δ
x
f
X,Y
(
s,t
)
dsdt