Which curve to select for finding the CDF of a function of a continuous joint distribution?

I came across a question which required to find the CDF of a function of a continuous joint distribution: $$W=XY$$.

The following is the joint PDF:

$$f_{X,Y}(x,y)=\begin{cases}\frac{xy}{4000}&,\, 1\le x\le 3,40\le y\le 60 \\ 0&,\text{ otherwise }\end{cases}$$

Therefore, $$40\le w\le 180$$.

The level plot of w with the constraints above on it

As you can see clearly the shape of the required area is changing by varying $$w$$, hence I can't find a single integral calculating the area for all the above values of w. So which curve should I select for doing the CDF?

Or do I need to define different CDF for different values of $$w$$, clubbing the values of w together for which the shape of the required area is similar?

• Please use MathJax for typesetting math. And I believe this question of yours is related to this one. – StubbornAtom Mar 24 at 11:20

If$$(X,Y)\sim \frac{xy}{4000}\,\Bbb I_{0\le x\le 3}\Bbb I_{40\le y\le 60}$$then \begin{align}(X,W)\sim &\frac{w}{4000}\,\Bbb I_{0\le x\le 3}\Bbb I_{40\le w/x\le 60}\overbrace{x^{-1}}^{\text{Jacobian}}\\ &=\frac{w}{4000 x}\,\Bbb I_{\max(0,w/60)\le x\le \min(3,w/40)}\Bbb I_{0\le w\le 180} \end{align} Hence $$\Bbb P(W\le w) = \int_0^{\min(180,w)}\frac{z}{4000}\int_{z/60}^{\min(3,z/40)}x^{-1}\text{d}x\text{d}z$$