Midterm 2 Solutions
EE126 - Fall 2000
Problem 1.
X Y
unif
(
;
1
1)
Z
=
XY
Let
U
=
X
and
V
=
XY
and form the jacobian for
f
UV
(
u v
).
f
UV
=
1
j
U
j
f
XY
(
u v=u
)
Marginalize with respect to u and employ the independence of X and Y.
f
V
(
v
)=
f
Z
(
z
)=
Z
+
1
;1
1
j
U
j
f
X
(
u
)
f
Y
(
z=u
)
du
f
Z
(
z
)=
Z
+
1
;1
1
2
j
U
j
f
Y
(
z=u
)
du
The domain for
f
Y
is bounded awayfrom 0bythe
z=u
term.
First consider
z >
0 for
f
Y
(
z=u
)
6
=0,1
>
j
u
j
>z
. Therefore, the integral
above can be completed for
z>
0by splitting it up into when
z<u<
1 and
;
1
<u<
;
z
.
so, for
z>
0
f
Z
(
z
)=
Z
1
z
1
4
U
du
+
Z
;
z
;
1
1
;
4
U
du
f
Z
(
z
)=
Z
1
z
1
2
U
du
f
Z
(
z
)=
;
1
2
ln
(
z
)
now, similarly,for
z<
0
f
Z
(
z
)=
Z
;
z
;
1
1
4
U
du
+
Z
1
z
1
;
4
U
du
f
Z
(
z
)=
Z
1
z
1
;
2
U
du
1
