MULTIVARIATE DISTRIBUTIONS
An
n
dimensional random vector
x
= [
x
1
, x
2
, . . . , x
n
] is an ordered set of
n
random
variables, each of which describes some aspect of a statistical outcome. We write
x
∈ R
n
to signify that
x
is a point in a real
n
dimensional space.
A function
f
(
x
) =
f
(
x
1
, x
2
, . . . , x
n
) that assigns a probability measure to every
point in
R
n
is called a multivariate p.d.f.
Consider the partitioned vector
x
=
x
1
x
2
.
To conserve space, we prefer to write this as a transposed row vector
x
= [
x
1
, x
2
] ,
where
x
1
= [
x
1
, x
2
, . . . , x
m
] and
x
2
= [
x
m
+1
, x
m
+2
, . . . , x
n
] . (Notice that, in this
notation, the subvector
x
1
and the leading scalar element
x
1
of the vector
x
are
represented by the same symbols. However, these entities can be distinguished by
their context.)
The marginal p.d.f, of
x
1
is
f
(
x
1
) =
x
2
f
(
x
1
, x
2
)
dx
2
=
x
n
· · ·
x
n
+1
f
(
x
1
, x
2
, . . . , x
m
, x
m
+1
,
· · ·
, x
n
)
dx
m
+1
, . . . , d
n
,
whereas the conditional p.d.f of
x
1
given
x
2
is
f
(
x
1

x
2
) =
f
(
x
)
f
(
x
2
)
=
f
(
x
1
, x
2
)
f
(
x
2
)
.
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.
 Spring '12
 D.S.G.Pollock
 Variance, Probability theory, Xn

Click to edit the document details