Cauchy
. Here
f
(
x
) =
1
π
1
1 + (
x

θ
)
2
.
What is interesting about the Cauchy is that it does not have ﬁnite mean, that is,
E

X

=
∞
.
Often it is important to be able to compute the density of
Y
=
g
(
X
). Let us give a couple of examples.
If
X
is uniform on (0
,
1] and
Y
=

log
X
, then
Y >
0. If
x >
0,
F
Y
(
x
) =
P
(
Y
≤
x
) =
P
(

log
X
≤
x
) =
P
(log
X
≥ 
x
) =
P
(
X
≥
e

x
) = 1

P
(
X
≤
e

x
) = 1

F
X
(
e

x
)
.
Taking the derivative,
f
Y
(
x
) =
d
dx
F
Y
(
x
) =

f
X
(
e

x
)(

e

x
)
,
using the chain rule. Since
f
X
= 1, this gives
f
Y
(
x
) =
e

x
, or
Y
is exponential with parameter 1.
For another example, suppose
X
is
N
(0
,
1) and
Y
=
X
2
. Then
F
Y
(
x
) =
P
(
Y
≤
x
) =
P
(
X
2
≤
x
) =
P
(

√
x
≤
X
≤
√
x
)
=
P
(
X
≤
√
x
)

P
(
X
≤ 
√
x
) =
F
X
(
√
x
)

F
X
(

√
x
)
.
Taking the derivative and using the chain rule,
f
Y
(
x
) =
d
dx
F
Y
(
x
) =
f
X
(
√
x
)
±
1
2
√
x
²

f
X
(

√
x
)
±

1
2
√
x
²
.
Remembering that
f
X
(
t
) =
1
√
2
π
e

t
2
/
2
and doing some algebra, we end up with
f
Y
(
x
) =
1
√
2
π
x

1
/
2
e

x/
2
,
which is a Gamma with parameters
1
2
and
1
2
. (This is also a
χ
2
with one degree of freedom.)
10. Multivariate distributions.
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 Spring '09
 wen
 Calculus, Chain Rule, Derivative, FY, dy dx

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