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My question is about how to get $\int^u_0\int^{u-y_2}_0$ and $\int^1_{u-1}\int^1_{u-y_2}$. How to find the sum of two uniform random variables by method of distribution? (without using convolution)

Let $(Y_1,Y_2)$ denote a random sample of size $n = 2$ from the uniform distribution on the interval $(0, 1)$. Find the probability density function for $U =Y_1 +Y_2$.

For 0 ≤ u ≤ 1,

$$F_U(u) = \iint _{y_1+y_2≤u}f (y_1, y_2) \, dy_1 \, dy_2 = \int^u_0\int^{u-y_2}_0(1)\,dy_1 \, dy_2 $$

For $1 < u ≤2$, (using complement)

$$F_U(u) = 1-\iint _{y_1+y_2≤u}f (y_1, y_2) \, dy_1 \, dy_2 = 1-\int^1_{u-1}\int^1_{u-y_2}(1) \, dy_1 \, dy_2 $$

I always use to sketch the graph to find the lower limit and upper limit of the integrals. But for this problem, I cannot figure out how to get $\int^u_0\int^{u-y_2}_0$ and $\int^1_{u-1}\int^1_{u-y_2}$.

Hope that someone can explain in logic and geometrically.

Michael Hardy
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AH Keng
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    This has been asked many times before. Did you search the site? – StubbornAtom Jun 24 '18 at 16:37
  • My problem is how to find the lower limit and upper limit of the double integral by using cumulative distribution technique(or method of distribution). The other problems are using convolutions to solve this problem. – AH Keng Jun 24 '18 at 16:44
  • You asked for the underlying geometry, which has probably been explained before. Once the geometry is clear, you could figure out the bounds. – StubbornAtom Jun 24 '18 at 17:00
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    Welcome to the site. This Q&A is pretty close to a duplicate. Maybe you will find the answer useful even though the 'underlying geometry' isn't on full display. // It is indeed worth trying to search this site before asking a question, but math notation is not easy to search and not everybody writes useful question titles, so you won't always succeed. – BruceET Jun 24 '18 at 21:32

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