Prob. & Stat. Lecture07 - functions of RVs
(cwliu@twins.ee.nctu.edu.tw)
7-4
1-1 Transformations of Variables
• Suppose that
X
1
and
X
2
are two
discrete
RVs with probability
function
f
(
x
1
,
x
2
).
• Suppose further that
Y
1
=
u
1
(
X
1
,
X
2
)
and
Y
2
=
u
2
(
X
1
,
X
2
)
define a
1-1 transformation between the set of points
(
x
1
,
x
2
)
and
(
y
1
,
y
2
).
.
• For a 1-1 transformation, the point
(
y
1
,
y
2
)
is related to one,
and only one, point
(
x
1
,
x
2
).
And the values
x
1
=
w
1
(
y
1
,
y
2
)
and
x
2
=
w
2
(
y
1
,
y
2
)
are the unique solution for the linear equation
y
1
=
u
1
(
x
1
,
x
2
)
and
y
2
=
u
2
(
x
1
,
x
2
)
•T
h
e
f
u
n
c
t
i
o
n
w
1
and
w
2
can be considered the inverse function
of
u
1
and
u
2
,
respectively
• The joint probability distribution of
Y
1
and
Y
2
is
)]
,
(
),
,
(
[
)]
,
(
),
,
(
[
)
,
(
)
,
(
2
1
2
2
1
1
2
1
2
2
2
1
1
1
2
2
1
1
2
1
y
y
w
y
y
w
f
y
y
w
X
y
y
w
X
P
y
Y
y
Y
P
y
y
g
=
=
=
=
=
=
=