Chapter 3:
Multivariate Random Variables
Joan Llull
Probability and Statistics.
QEM Erasmus Mundus Master. Fall 2015
joan.llull [at] movebarcelona [dot] eu
Joint and Marginal Distributions
Chapter 3. Fall 2015
2
Joint and marginal cdfs
Multivariate random variable
: a vector that includes several
(scalar) random variables:
X
= (
X
1
,...,X
K
)
0
.
Joint cdf
:
F
X
1
...X
K
(
x
1
,...,x
K
)
≡
P
(
X
1
≤
x
1
,X
2
≤
x
2
K
≤
x
K
)
Marginal cdf
:
F
i
(
x
)
≡
P
(
X
i
≤
x
) =
P
(
X
1
≤ ∞
i
≤
x,.
..,X
K
≤ ∞
)
=
F
(
∞
,...,x,.
..,
∞
)
.
Chapter 3. Fall 2015
3
Joint pmfs and pdfs
Joint pmf
(discrete var.):
P
(
X
1
=
x
1
,X
2
=
x
2
,...,X
K
=
x
K
)
.
Joint pdf
(continuous var.): a joint pdf
f
X
1
...X
K
(
z
1
,...,z
K
)
satis es:
F
X
1
...X
K
(
x
1
,...,x
K
)
≡
Z
x
1
∞
...
Z
x
K
∞
f
X
1
...X
K
(
z
1
K
)
dz
1
...dz
K
.
Properties of joint pdfs
:
f
X
1
...X
K
(
x
1
K
)
≥
0
for all
x
1
K
.
F
X
1
...X
K
(
∞
,...,
∞
) =
R
∞
∞
...
R
∞
∞
f
X
1
...X
K
(
z
1
K
)
dz
1
...dz
K
= 1
.
P
(
a
1
≤
X
1
≤
b
1
,...,a
K
≤
X
K
≤
b
k
) =
R
b
1
a
1
...
R
b
K
a
K
f
(
z
1
K
)
dz
1
...dz
K
.
P
(
X
1
=
a
1
K
=
a
K
) = 0
.
P
(
X
1
=
a,a
2
≤
X
2
≤
b
2
K
≤
X
K
≤
b
K
) = 0
.
(examples with discrete and continuous variables)
Chapter 3. Fall 2015
4
Marginal pmfs and pdfs
Marginal pmf
(discrete var.):
P
(
X
i
=
x
)
≡
X
x
1
...
X
x
K
P
(
X
1
=
x
1
,...X
i
=
x,.
..,X
K
=
x
K
)
.
Marginal pdf
(continuous var.):
f
i
(
x
) =
Z
∞
∞
...
Z
∞
∞
f
X
1
...X
K
(
z
1
,...,x,.
..,z
K
)
dz
1
...dx
i

1
dx
i
+1
...dz
K
,
or equivalently:
F
i
(
x
) =
Z
x
∞
f
i
(
z
)
dz.
(examples with discrete and continuous variables)
Chapter 3. Fall 2015
5
Conditional Distributions and
Independence
Chapter 3. Fall 2015
6
Conditional probability and independence
Let
A
,
B ⊂ F
. The probability that
A
occurs
given that
B
occurred, denoted by
P
(
AB
)
is formally de ned as:
P
(
)
≡
P
(
A∩B
)
P
(
B
)
.
Bayes' rule
:
P
(
) =
P
(
)
P
(
B
) =
P
(
BA
)
P
(
A
)
⇒
P
(
) =
P
(
)
P
(
B
)
P
(
A
)
.
A
and
B
are
independent
if (the three below are equivalent):
P
(
) =
P
(
A
)
P
(
) =
P
(
B
)
P
(
) =
P
(
A
)
P
(
B
)
.
Chapter 3. Fall 2015
7
Conditional cdfs
Let
X
be a random variable, and
A
an event, with
P
(
A
)
6
= 0
. The
conditional cdf
of
X
given
A
occurred is:
F
X
A
(
x
)
≡
P
(
X
≤
x
) =
P
(
X
≤
x
∩ A
)
P
(
A
)
.
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