Chapter 6 summary
Jointly Distributed Random Variables
•
Basic problem:
Let
X
and
Y
be random variables. Consider the pair (
X,Y
). This is not a random
variable because it isn’t a real-valued function (since its values are in
R
2
, not
R
). However, if
D
is any
subset of
R
2
, we’d like to know
P
((
X,Y
)
∈
D
), i.e. the probability that (
X,Y
) takes values in
D
. This
is achieved in case
X
and
Y
are discrete by using the
joint probability mass function
p
X,Y
(
x,y
)
and in case
X
and
Y
are continuous by using the
joint probability density function
f
X,Y
(
x,y
).
These are each described next.
•
Joint probability mass function:
If
X
and
Y
are discrete random variables, the
joint probability
mass function
is deﬁned to be
p
X,Y
(
x,y
) =
P
(
X
=
x
and
Y
=
y
)
.
For any subset
D
of the set of possible values of (
X,Y
),
P
((
X,Y
)
∈
D
) =
X
(
x,y
)
∈
D
p
X,Y
(
x,y
)
.
The joint mass function has the property that it is
≥
0 and
∑
x,y
p
X,Y
(
x,y
) = 1 if the sum is taken
over all possible pairs of values that (
X,Y
) can take.
•
Joint probability density function:
If
X
and
Y
are continuous random variables, the
joint prob-
ability density function
is denoted by
f
X,Y
(
x,y
). It has the property that it is
≥
0 and for subsets
D
of
R
2
for which we can deﬁne
P
((
X,Y
)
∈
D
) we have
P
((
X,Y
)
∈
D
) =
Z
D
Z
f
X,Y
(
x,y
)
dA.
The
support
of the joint probability density function is basically the set of (
x,y
) where
f
X,Y
(
x,y
)
>
0.
More precisely, a point is in the support of
f
X,Y
if for every disk centered at that point, the probability
that (
X,Y
) is in that disk is strictly positive. In writing the above double integral as an iterated
double integral, it is important to draw the region bounded by the intersection of