The marginal density is obtained by integrating f
W Z
(w, z) over the shaded region, D. For convenience, let
f
1
= y = z − w
f
2
= 2 − y = 2 −z + w.
Then,
f
Z
(z) =
D
f
W Z
(w, z) dw
=
z
0
f
1
dw 0 ≤ z < 1
1
z−1
f
1
dw +
z−1
0
f
2
dw 1 ≤ z < 2
1
z−2
f
2
dw 2 ≤ z < 3
That is,
f
Z
(z) =
z
2
/2, 0 ≤ z < 1
−z
2
+ 3z − 3/2 1 ≤ z < 2
z
2
/2 − 3z + 9/2 2 ≤ z < 3
0 elsewhere
.
106