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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 WEBER
 Probability theory, continuous random vectors

Click to edit the document details