while finding PDF of $W=X+Y$ from given Joint PDF $f_{X,Y}(x,y)$ How to find the limits of integral? RV $X$ and $Y$ have joint PDF:
$$f_{X,Y}(x,y)=
\begin{cases}
8xy & 0\le y \le x \le1 \\
0 & \text{otherwise}
\end{cases}$$
Find PDF of W=X+Y
I know that I need to use : $f_W(w)=\int_{-\infty}^\infty f_{X,Y}(x,w-y) \, dx$. But im confused about limits. I saw some examples but couldnt figure out how they choose limits. (obviously not $-\infty$ to $\infty$) and it seems not $(0$ to $y$ and $y$ to $x)$. Can anyone explain it in simple way?
sorry for my bad english but i think i hit the point
 A: You wrote $$0\le y\le x\le 1.\tag1$$  That is how you choose limits.  Suppose you want to integrate the density over its whole domain and get $1$.  You have
$$
\int_0^1 \cdots\cdots \,dx
$$
and inside that you have an integral with respect to $y$.  According to $(1)$, $y$ must remain between $0$ and $x$, so you have
$$
\int_0^1 \left( \int_0^x\cdots\cdots\,dy \right) \,dx.
$$
Alternatively, you can put the integral with respect to $y$ on the outside, so you have
$$
\int_0^1 \cdots\cdots\,dy.
$$
Inside that you integrate with respect to $x$.  According to $(1)$, $x$ runs from $y$ to $1$, so you have
$$
\int_0^1\left( \int_y^1 \cdots\cdots\,dx \right)\,dy.
$$
A: Express the joint density function using indicators, substitute $y=w-x$, rearrange the indicators to give the bounds on $x$ relative to $w$, and then integrate with respect to $x$ over those bounds to give the density function for $W = X+Y$.
$$\begin{align}
f(x, y) & = 8 xy \;\mathbf 1_{0\le y\le x\le 1}
\\[2ex]
f(x, w-x) & = 8 x(w-x) \;\mathbf 1_{0\le (w-x)\le x\le 1} & y = w- x
\\ & = 8 (wx-x^2) \;\mathbf 1_{0\le x\le w\le 2 x \le 2}
\\ & = 8 (wx-x^2) \;\mathbf 1_{0\le w\le 2}\;\mathbf 1_{w/2\le x \le \min(w,1)}
\\[2ex]
 f_{\small W}(w) & = \mathbf 1_{0\le w\le 1}\int_{w/2}^w 8(wx-x^2)\operatorname d x 
+ \mathbf 1_{1 < w\le 2}\int_{w/2}^1 8(wx-x^2)\operatorname d x
\\[1ex]
 & = \frac{2w^3}{3}\mathbf 1_{0\le w\le 1} + \frac{12w -8-2w^3}{3}
\mathbf 1_{1\le w\le 2}
\end{align}$$
