# Find the probability density function of sum of two marginal probability density functions

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!

• The marginals you calculated look "suspicious". They can take positive values outside $[0,1]$ while $(X,Y)$ has a subset of $[0,1]^2$ as support. There is definitely something wrong with your calculation. Fortunately you do not need them for calculating $f_W(w)$. – drhab Aug 12 at 15:09
• Made a typo in the marginals, they should be correct now. – xbys Aug 12 at 15:13
• Not completely. Also $0\leq x\leq 2$ must be changed into $0\leq x\leq 1$. Same story of $y$. – drhab Aug 12 at 15:15
• You're right, they should be complete now . – xbys Aug 12 at 15:17

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 $$w\notin[0,1]$$.