Finding probability density function using CDF method I'm working on the following problem, but I'm not really sure how to approach it - it's different from anything I've seen before!
The problem is as follows: 
Consider the probability density function

$f_{X,Y}(x,y) = \left\{\frac{8+xy^3}{64}\right\}$ if $-1<x<1, -2<y<2$,
  with probabilility $0$ otherwise.

What I'm trying to do is find the PDF of $W=2X+Y$, which is causing me some trouble - in fact I hardly know where to start! So I know the support of X is $-1<x<1$ and the support of Y is $-2<y<2$, since the region is a square. I think this means that the support of W is $-3<w<3$, since $W=2X+Y$. 
This is where I start to get confused. I believe in order to find the PDF, I first want to find the CDF of W, and then take the derivative of that. In order to find the CDF, I want to  evaluate a double integral in terms of X and Y with the given PDF. However, I don't know what to set the bounds of these integrals to! In fact, I'm not really sure how to even begin; I feel like it might involve solving for X and Y in terms of W $(y=2x-w)$ and $(x=\frac{y-w}{2})$ but I don't know exactly what (if anything) to do with these!
Thank you so much for your help - I really appreciate it!
Sarah
 A: Begin by drawing a sketch of the $x$-$y$ plane and marking on it the region where $f_{X,Y}$ is nonzero. (Hint: it is not a square). 
STOP. Do not proceed if you have ignored the previous statement and
have not made a sketch. Reading beyond this point is useless.
You are not correct when you say that $W \in (-3,3)$. $W$ can take on values in
$(-4,4)$ since $2\times 1 + 2 = 4$, , So you know already that $F_W(w)=0$ for $w-<4$ and $1$ for $W\geq 4$. Now, pick a fixed number $w \in(-4,4)$, and draw the line $2x+y=w$. Then, $W=2X+Y\leq w$ if the point $(X,Y)$ lies on or below this line. Can you integrate to find $P(W\leq w)=F_W(w)$ for your chosen value of $w$? (Hint: you may need to break up the integral into two or more parts to carry out the calculation). Lather, rinse, repeat for different choices of $w$ which will give you different lines and different
integrals to compute. You may be able to see a trend developing and ultimately
be able to write a complete expression for $F_W(w)$ like
$$F_W(w) = \begin{cases}0, & w < -4,\\
\cdots,& -4 < w < ?,\\
\cdots,& ? < w <??\\
\cdots & \cdots\\
1, & w \geq 4,\end{cases}$$
after just a few tries with different values of $w$.
Then, differentiate to find the pdf $f_W(w)$ and you are done.
