We suppose that there is a probability measure defined over
R
2
such that, for
any
A ⊂ R
2
,
P
(
A
) is the probability that (
x, y
) falls in
A
. Thus, for example, if
A
=
{
a < x
≤
b, a < x
≤
b
}
, which is a rectangle in the plane, then
P
(
A
) =
d
y
=
c
b
x
=
a
f
(
x, y
)
dx
dy.
This is a double integral, which is performed in respect of the two variables in
succession and in either order. Usually, the braces are omitted, which is allowable
if care is taken to ensure the correct correspondence between the integral signs and
the differentials.
Example.
Let (
x, y
) be a random vector with a p.d.f of
f
(
x, y
) =
1
8
(6
−
x
−
y
);
0
≤
x
≤
2;
2
≤
y
≤
4
.
It needs to be confirmed that this does integrate to unity over the specified range
of (
x, y
). There is
1
8
2
x
=0
4
y
=2
(6
−
x
−
y
)
dydx
=
1
8
2
x
=0
6
y
−
xy
−
y
2
2
4
2
dx
=
1
8
2
x
=0
(6
−
2
x
)
dx
=
1
8
6
x
−
x
2
2
0
=
8
8
= 1
.
Moments of a bivariate distribution.
Let (
x, y
) have the p.d.f.
f
(
x, y
). Then,
the expected value of
x
is defined by
E
(
x
) =
x
y
xf
(
x, y
)
dydx
=
x
xf
(
x
)
dx,
if
x
is continuous, and by
E
(
x
) =
x
y
xf
(
x, y
) =
x
xf
(
x
)
,
if
x
is discrete.
Joint moments of
x
and
y
can also be defined. For example, there is
x
y
(
x
−
a
)
r
(
y
−
b
)
s
f
(
x, y
)
dydx,
where
r, s
are integers and
a, b
are fixed data. The most important joint moment
for present purposes is the covariance of
x
and
y
, defined by
C
(
x, y
) =
x
y
{
x
−
E
(
x
)
}{
y
−
E
(
y
)
}
f
(
x, y
)
dydx.
If
x, y
are statistically independent, such that
f
(
x, y
) =
f
(
x
)
f
(
y
)
, then
their joint moments can be expressed as the products of their separate
moments.
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