Suppose that
has pdf
. What is the pdf of
? Or
if
, what is the pdf of
This is a special case of asking for the pdf of
when
 |
(4.43) |
for a one-to-one transformation
:
. Define the Jacobian of
as
and let
be the absolute value of the determinant
of this Jacobian. The pdf of
is given by
 |
(4.44) |
Using this we can answer the introductory questions, namely
with
and hence
.
So the pdf of
is
.
This introductory example is a special case of
The inverse transformation is
Therefore
and hence
 |
(4.45) |
EXAMPLE 4.12
Consider

with density

,
Then
and
Hence
EXAMPLE 4.13
Consider

with density

and

.
According to (
4.43)

and hence the Jacobian is
The pdf of

is therefore:
Summary

-
If
has pdf
, then a transformed random vector
, i.e.,
, has pdf
, where
denotes the Jacobian
.

-
In the case of a linear relation
the pdf's of
and
are related via
.