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Computing the pdf of
Y
=
T
(
X
)
.
Let
X
be continuous with pdf
f
X
.
•
Sometimes
Y
=
T
(
X
) is also continuous.
•
To compute the density of
Y
adjust by
the derivative of the inverse transfor
mation
.
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View Full Document Monotone transformations
:
the function
T
is
either increasing or decreasing on the range of
X
.
A monotone function
T
is automatically one
toone.
The pdf of
Y
:
f
Y
(
y
) =
f
X
±
T

1
(
y
)
²
³
³
³
³
³
dT

1
(
y
)
dy
³
³
³
³
³
.
Example
Let
X
be a standard exponential
random variable.
Compute the density of
Y
= 1
/X
.
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View Full Document This method sometimes works even when the
function
T
is not monotone.
•
The equation
T
(
x
) =
y
may have several roots,
T

1
1
(
y
),
T

1
2
(
y
)
,...
.
•
Each of the roots contributes to the den
sity of
Y
=
T
(
X
):
f
Y
(
y
) =
X
i
f
X
±
T

1
i
(
y
)
²
³
³
³
³
³
³
dT

1
i
(
y
)
dy
³
³
³
³
³
³
.
Example
Let
X
be standard normal, and
Y
=
T
(
X
), with
T
(
x
) =
(

x
if
x
≤
0
2
x
if
x >
0.
Compute the pdf of
Y
.
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View Full Document Transformations of random vectors
Statement of the problem:
let (
X
1
,...,X
k
)
be a random vector with a known joint pmf (if
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This note was uploaded on 12/03/2010 for the course OR&IE 3500 taught by Professor Samorodnitsky during the Fall '10 term at Cornell University (Engineering School).
 Fall '10
 SAMORODNITSKY

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