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I am trying to find the distribution of the sum of two random variables $(x_1,x_2)$ which have the following distribution:

$$f(x_1,x_2) = \begin{cases}2 ( x_1 + x_2) & 0 \le x_1 \le x_2 \le 1 \\ 0 & \text{otherwise} \end{cases}$$

The expression I thought would work was

TransformedDistribution[
 x1 + x2, {x1, x2} \[Distributed] 
  ProbabilityDistribution[2 (x1 + x2), {x1, 0, x2}, {x2, x1, 1}]]

But this only returns the expression unevaluated. What would be the proper method to calculate the result?

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I figured it out -- the expression is representing the distribution object. To get the formula, I had to call PDF on the distribution. The code would then look like

PDF[TransformedDistribution[
  x1 + x2, {x1, x2} \[Distributed] 
   ProbabilityDistribution[2 (x1 + x2), {x1, 0, x2}, {x2, x1, 1}]], w]

The result is

$$\begin{cases} -((-2+w)w) & 1 < w < 2 \\ w^2 & 0 <w \le 1 \\ 0 & \text{True}\end{cases}$$

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    $\begingroup$ Showing your result would be helpful. $\endgroup$
    – bbgodfrey
    Nov 14 '21 at 3:07

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