11. Incorrect. The
answer is false not true. It is possible to obtain such a formula by making
a change of variable: u = x + 2y
and v = x. The simultaneous solution is x
= v and y = 1/2(u - v). Thus,
after changing variables, the integration region changes from 0 < x < 4 and 1 < y < 5 (region E),
changes to 0 < v < 4 and 2 <
u - v < 10 (region D), which is shown shaded in the
following figure:
We have to apply Theorem 2-4,
p.45 in Schaum's Outlines Probability and Statistics.
Let x and y be continuous random variables having joint density function f(x,y,).
Let us define
u = φ1(x, y), v = φ2(x, y) where x = ψ1(u, v), y = ψ2(u, v). The joint density function of u
and v is given by g(u, v) where:
where J = Jacobian determinant (or simply Jacobian) that is given by:
.
The determinant of 2x2 matrix
is equal to: ad - cb. In the problem, the Jacobian is:
So, the change of variable allows us to obtain: