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
![\begin{displaymath}
X = u(Y)
\end{displaymath}](mvahtmlimg1341.gif) |
(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
![\begin{displaymath}
f_Y(y) = \abs(\vert{\cal J}\vert) \cdot f_X\{u(y)\}.
\end{displaymath}](mvahtmlimg1346.gif) |
(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
![\begin{displaymath}
f_Y(y) = \abs(\vert\data{A}\vert^{-1})f_X\{\data{A}^{-1}(y-b)\}.
\end{displaymath}](mvahtmlimg1354.gif) |
(4.45) |
EXAMPLE 4.12
Consider
![$X=(X_{1},X_{2})\in\mathbb{R}^2$](mvahtmlimg1355.gif)
with density
![$f_X(x)=f_{X}(x_{1},x_{2})$](mvahtmlimg1356.gif)
,
Then
and
Hence
EXAMPLE 4.13
Consider
![$X\in\mathbb{R}^1$](mvahtmlimg1269.gif)
with density
![$f_X(x)$](mvahtmlimg1364.gif)
and
![$Y=\exp(X)$](mvahtmlimg1365.gif)
.
According to (
4.43)
![$x=u(y)=\log(y)$](mvahtmlimg1366.gif)
and hence the Jacobian is
The pdf of
![$Y$](mvahtmlimg237.gif)
is therefore:
Summary
![$\ast$](mvahtmlimg108.gif)
-
If
has pdf
, then a transformed random vector
, i.e.,
, has pdf
, where
denotes the Jacobian
.
![$\ast$](mvahtmlimg108.gif)
-
In the case of a linear relation
the pdf's of
and
are related via
.