AMS 311
Fall Semester, 2010
Chapter Six: Jointly Distributed Random Variables
6.7. Joint Probability Distribution of Functions of Random Variables
Transformation of two random variables is a crucial problem and hard to handle. It is
important to review your multivariable calculus so that you are up to speed technically.
The probability theory is given in this section and may look hard. The calculations are
straightforward but require careful attention.
Let
1
X
and
2
X
be jointly continuous random variables with joint probability density
function
).
,
(
2
1
,
2
1
x
x
f
X
X
In the discussion below, I will refer to
1
X
as
1
old
and
2
X
as
.
2
old
Let
)
,
(
2
1
1
1
X
X
g
Y
=
and
).
,
(
2
1
2
2
X
X
g
Y
=
I will refer to
1
Y
as
1
new
and
2
Y
as
.
2
new
Assume that the functions
)
,
(
2
1
1
1
x
x
g
y
=
and
)
,
(
2
1
2
2
x
x
g
y
=
can be uniquely solved for
1
x
and
2
x
. Further assume that
0
)
det(
)
,
(
2
2
1
2
2
1
1
1
2
1
≠
∂
∂
∂
∂
∂
∂
∂
∂
=
x
g
x
g
x
g
x
g
x
x
J
at all points
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 Spring '08
 Tucker,A
 Normal Distribution, Probability theory, probability density function, Cumulative distribution function

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