
The marginal density is given by
f
Z
(z) =
0.5
2z
f
W Z
(w, z) dw +
−2z
−0.5
f
W Z
(w, z) dw
=
0.5
2z
f
XY
(x = w, y = z/x) |J|dw +
−2z
−0.5
f
XY
(x = w, y = z/x) |J|dw
=
0.5
2z
−
1
w
dw +
−2z
−0.5
−
1
w
dw
where
−
1
w
=
1
w
for w > 0
−
1
w
= −
1
w
for w < 0.
Then,
f
Z
(z) =
0.5
2z
1
w
dw +
−2z
−0.5
−
1
w
dw
= 2 ln(0.5) − 2 ln (2z)
= −4 ln(2) − 2 ln(z) for 0 < z < 1/4.
For z < 0, the marginal density is given by
f
Z
(z) =
0.5
−2z
f
W Z
(w, z) dw +
2z
−0.5
f
W Z
(w, z) dw
=
0.5
−2z
−
1
w
dw +
2z
−0.5
−
1
w
dw
= 2 ln(0.5) − 2 ln (2 |z|) for −1/4 < z < 0.
We can write
f
Z
(z) = 2 ln(0.5) − 2 ln (2 |z|) for −1/4 < z < 1/4.
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