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Suppose we choose independently two numbers $X$ and $Y$ , at random, from the interval $[0,1]$ with uniform probability density. What is the density of their sum $Z = X + Y$ ?

I know that the answer is: $$ f_Z(z) = \left\{ \begin{array}{lr} z & : 0 \leq z \leq 1\\ 2-z & : 1 < z \leq 2\\ 0 & : \text{otherwise} \end{array} \right.$$

But I'm having a hard time seeing the translation from the P.D.F. of just a single random variable to the sum of two.

I think an explanation for three random variables (i.e. $F_Z'(z')$ for $Z' = A + B + C$ where $A$,$B$, and $C$ are uniformly distributed on $[0,1]$) would help illustrate this.

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    This has been asked and answered many times. The three variable case is more complicated, and will not help the understanding. You should search for other answers. There will also be picture versions of the justification on MSE, the problem is very geometric. My answer happens to be a convolution-based one, I prefer the approach through the cumulative distribution function of the sum. – André Nicolas May 08 '13 at 18:40

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