Given two random variables $X$ and $Y$ with joint probability density function
$$
f_{X,Y}(x,y) =
\begin{cases}
2, & \text{if x $\ge$ 0, y $\ge$ 0 , $x + y$ $\le$ 1} \\
0, & \text{elsewhere}
\end{cases}
$$
let $W = X + Y$
My question is how I calculate the probabilty density function $f_W(w)$ from here?
If calculated the marginal probabilty density functions
$$f_x(x) = 2-2x \text{ for } 0 \le x \le 1$$
$$f_Y(y) = 2-2y \text{ for } 0 \le y \le 1$$
But i have no clue how to go from here to the probability density function $f_W(w)$
Any pointers or solutions are highly appreciated!
Thanks in advance!
Best Answer
For $w\geq0$ define triangle: $$\Delta_w=\{(x,y)\in[0,1]^2\mid x+y\leq w\}$$
Then $(X,Y)$ has uniform distribution on triangle $\Delta_1$
For $w\in[0,1]$ we find: $$F_W(w)=P(X+Y\leq w)=P((X,Y)\in\Delta_w)=2\lambda(\Delta_w)=w^2$$ where $\lambda$ denotes the Lebesgue measure.
The PDF of $W$ can be found as derivative of the CDF, i.e. for $w\in[0,1]$ we have:$$f_W(w)=2w$$ Further we can take $f_W(w)=0$ if $w\notin[0,1]$.