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**noob mathematician** If X and Y are independent r.v, both uniformly distributed on (0, 1).

Given that $\displaystyle f_{X+Y}(a)=\int_{-\infty}^{\infty}f_X(a-y)f_Y(y)dy$

we obtain $\displaystyle f_{X+Y}(a)=\int_{0}^{1}f_X(a-y)dy$

For $\displaystyle 0\leq a\leq 1$, we have $\displaystyle f_{X+Y}=\int_{0}^{a}dy=a$

For $\displaystyle 1<a<2$, we have $\displaystyle f_{X+Y}=\int_{a-1}^{1}dy=2-a$

How do I determine the two upper and lower bounds?

Such as for this case we are using: $\displaystyle \int_{0}^{a}$ and $\displaystyle \int_{a-1}^{1}$ ??

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