The joint pmf of (
X, Y
) contains all information about the distribution of
X
and
Y
. In particular, we can extract
f
X
and
f
Y
, the pmfs for
X
and
Y
individually, from
the joint pmf
f
XY
. The individual pmfs
f
X
and
f
Y
are referred to as the
marginal
probability mass functions
for
X
and
Y
. The marginal pmf of
X
evaluated at a
point
x
is obtained by summing
f
XY
(
x, y
) over all values of
y
, while the marginal
pmf of
Y
evaluated at a point
y
is obtained by summing
f
XY
(
x, y
) over all values
of
x
:
f
X
(
x
) =
X
y
f
XY
(
x, y
) and
f
Y
(
y
) =
X
x
f
XY
(
x, y
)
.
A pair of random variables (
X, Y
) is said to be continuous if the probability
of (
X, Y
) being equal to any fixed pair of values (
x, y
) is zero. In fact, if we want
to be very precise, then continuity means a little more than this: the probability
of (
X, Y
) lying in any subset of the plane that has zero area must be zero. When
(
X, Y
) is continuous, its behavior may be described by its
joint probability density
function
, or joint pdf, denoted
f
XY
. If (
a, b
) and (
c, d
) are two intervals on the real
line, then the probability that (
X, Y
) lies in the rectangle (
a, b
)
×
(
c, d
) – that is,
the probability that
X
lies in (
a, b
) and
Y
lies in (
c, d
) – is equal to the volume
underneath
f
X
within the rectangle (
a, b
)
×
(
c, d
):
P
(
a < X < b
and
c < Y < d
) =
Z
d
c
Z
b
a
f
XY
(
x, y
)d
x
d
y.
Clearly, we must have
f
XY
(
x, y
)
≥
0 for all
x
and
y
, and the total volume under-
neath the joint pdf must be one:
R
∞
-∞
R
∞
-∞
f
XY
(
x, y
)d
x
d
y
= 1.
As in the discrete case, we may extract the pdfs
f
X
and
f
Y
for
X
and
Y
from
the joint pdf
f
XY
for (
X, Y
). The univariate pdfs
f
X
and
f
Y
are referred to as the
marginal probability density functions
, and may be obtained by “integrating out”
one of the two arguments of
f
XY
:
f
X
(
x
) =
Z
∞
-∞
f
XY
(
x, y
)d
y
and
f
Y
(
y
) =
Z
∞
-∞
f
XY
(
x, y
)d
x.
6
Covariance and independence
Given a discrete pair of random variables (
X, Y
) with joint pmf
f
XY
, and a real
valued function
g
of two variables, we may compute the expected value of
g
(
X, Y
)
as follows:
E
(
g
(
X, Y
)) =
X
x
X
y
g
(
x, y
)
f
XY
(
x, y
)
.
Similarly, if (
X, Y
) is continuous with joint pdf
f
XY
, we have
E
(
g
(
X, Y
)) =
Z
∞
-∞
Z
∞
-∞
g
(
x, y
)
f
XY
(
x, y
)d
x
d
y.
6