
Section 4.2: Functions of Two or More RVs
4.13. For the function Z = XY, find f
Z
(z) and sketch for the cases:
(a) f
XY
(x, y) = [(b −a)(d −c)]
−1
.
(b) f
XY
(x, y) = exp[−(x + y)].
Solution: Using the general method, we define a variable W,
W = X.
Solving for X and Y ,
X = W = g
1
Y =
Z
W
= g
2
.
The Jacobian is given by
J =
∂g
1
/∂z ∂g
1
/∂w
∂g
2
/∂z ∂g
2
/∂w
=
0 1
1/W −Z/W
2
= −
1
W
.
f
XY
(x, y) is defined on the rectangle defined by a < X < b and c < Y < d, Since X = W, we know
a < W < b. The range for Z can be derived from that of Y. It is given that c < Y < d. Writing Y in terms
of Z and W,
c < Y < d → c <
Z
W
< d.
Solving for Z,
cW < Z < dW.
Z ranges between two lines: Z