Multivariate (actually mostly bivariate!) Distributions
DISCRETE:
Let
X
and
Y
be discrete random variables. Then the
joint pmf
of
X
and
Y
is
p
X,Y
(
x,y
) =
P
(
X
=
x,Y
=
y
)
.
[Note: Read
P
(
X
=
x,Y
=
y
) as
P
(
X
=
x
and
Y
=
y
).]
An important property that all joint pmf’s should satisfy is :
X
x
X
y
p
X,Y
(
x,y
) = 1
.
The
marginal pmf
of
X
is
p
X
(
x
) =
X
y
p
X,Y
(
x,y
) =
P
(
X
=
x
)
and the
marginal pmf
of
Y
is
p
Y
(
y
) =
X
x
p
X,Y
(
x,y
) =
P
(
Y
=
y
)
.
The following are the respective
conditional pmf’s
p
X

Y
(
x

y
) =
P
(
X
=
x

Y
=
y
) =
P
(
X
=
x,Y
=
y
)
P
(
Y
=
y
)
=
p
X,Y
(
x,y
)
p
Y
(
y
)
p
Y

X
(
y

x
) =
P
(
Y
=
y

X
=
x
) =
P
(
X
=
x,Y
=
y
)
P
(
X
=
x
)
=
p
X,Y
(
x,y
)
p
X
(
x
)
Let
k
(
X,Y
) be a function of the random variables, then the
expected value
(or mean or average
value) of
k
(
X,Y
) is
E
[
k
(
X,Y
)] =
X
x
X
y
k
(
x,y
)
p
X,Y
(
x,y
)
.
The
covariance
of the random variables
X
and
Y
, often written as
Cov
(
X,Y
) or
σ
XY
, and is given
by
Cov
(
X,Y
) =
E
[(
X

E
(
X
))(
Y

E
(
Y
))]
=
E
[
XY
]

E
[
X
]
E
[
Y
]
.
If