Let X~UNIF(0,1) and Y~UNIF(0,1)
then I know that X+X~UNIF(0,2)
but how will X+Y be distributed?
Is there an easy way to determine this? I think it should be a triangle distribution of some sort.
If you look at the discrete uniform and think of 2 dice. If you throw the 2 dice there is 1 possible way to get 2. There are 2 ways to get 3. 3 ways to get 4 etc. up to 1 way to get 12. But how will this be adjusted for continuous uniform?
By X+X you mean 2X, well that is U(0,2).
For W=X+Y, wher you need independence, the classic way to do is solve via the CDF of W.
Draw the unit square (0,1)X(0,1) and find F(w) where 0<w<1 and 1<w<2.
The derivative will give you the density which is
f(w)=w on (0,1) and f=2-w on [1,2) and 0 elsewhere.