ECE 6010
Lecture 4 – Change of Variables
Changing variables: One dimension
A simply invertible function
Let
Y
=
g
(
X
)
, where
X
is a continuous r.v.
and
g
is a onetoone, onto, measurable
function. Then
F
Y
(
y
) =
P
(
Y
≤
y
) =
P
(
g
(
X
)
≤
y
) =
P
(
X
≤
g

1
(
y
)) =
F
X
(
g

1
(
y
))
So we can determine the distribution of
Y
. Let us now take a different point of view that
will allow us to generalize to higher dimensions and develop and understand a commonly
used formula.
Consider an interval along the
X
axis,
P
(
x
≤
X
≤
x
+
dx
)
≈
f
X
(
x
)
dx
Suppose the function
g
(
x
)
has a positive derivative. The interval along the
Y
axis, when
Y
=
g
(
X
)
is
dy
≈
dy
dx
dx
at the point
x
. The probability that
X
falls in its interval is the
probability that
Y
falls in its interval:
P
(
x
≤
X
≤
x
+
dx
) =
P
(
y
≤
Y
≤
y
+
dy
)
where
y
=
g
(
x
)
, or equivalently,
x
=
g

1
(
y
)
. Then
f
X
(
x
)
dx
≈
f
Y
(
y
)
dy
That is
f
Y
(
y
) =
f
X
(
x
)
dx
dy
±
±
±
±
x
=
g

1
(
y
)
=
f
X
(
g

1
(
y
))
dx
dy
.
If we take the other case that
g
(
x
)
has a negative derivative, we have to take
f
X
(
x
)
dx
=
f
Y
(
y
)(

dy
)
. Combining these together we obtain
f
Y
(
y
) =
f
X
(
g

1
(
y
))
±
±
±
±
dx
dy
±
±
±
±
=
f
X
(
g

1
(
y
))

dy/dx

=
f
X
(
g

1
(
y
)

g
0
(
g

1
(
x
))
.
Example 1
Let
Y
=
aX
+
b
. Then
f
Y
(
y
) =
1

a

f
X
((
y

b
)
/a
)
.
2
Example 2
Suppose
f
X
(
x
) =
a/π
x
2
+
a
2
(Cauchy). Let
Y
= 1
/X
. Then
f
Y
(
y
) =
1
/aπ
y
2
+ 1
/a
2
(Cauchy)
2
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2
Example 3
Suppose
X
∼ U
(
a, b
)
, with
0
< a < b
. Then
f
X
(
x
) =
1
b

a
for
x
∈
[
a, b
]
. Let
Y
= 1
/X
. Then
f
Y
(
y
) =
1
(
b

a
)
y
2
for
1
b
< y <
1
a
.
2
Example 4
Let
Y
=
e
X
. Then
f
Y
(
Y
) =
1
y
f
X
(ln
y
)
y >
0
.
If
X
∼ N
(
μ, σ
2
)
then
f
Y
(
y
) =
1
yσ
√
2
π
e

(ln
y

μ
)
2
/
2
σ
2
This density is
lognormal.
2
Example 5
Suppose
y
=
g
(
x
) = tan
x
, or
x
= tan

1
y
. This has an infinite number of
solutions. However, if we take
x
∈
(

π/
2
, π/
2)
, then there is a unique inverse. We have
g
0
(
x
) =
1
cos
2
x
= 1 +
y
2
.
Now let
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 Spring '08
 Stites,M
 Cartesian Coordinate System, Inverse function

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