Easy: X is defined by a uniform distribution over [0,1]; Y is defined by a Dirac delta distribution at 0.
...or more simply: X and Y are both defined by a uniform distribution over [0,1/2]. What are you really trying to ask here?
The first solution is technically correct, but it wasn't the one I had in mind since Y is a constant (thus making it a trivial solution).The second solution is incorrect. When you add two random variables together, the sum is a random variable whose density is a convolution of the two original random variables. In the case of two uniform random variables over [0,1/2], the sum is a random variable whose density looks like a symmetric triangle over [0,1] and centered at 1/2.
Ok, so I was tired and forgot about dice. :)
How about a coinflip choice of 0 or .5 and a uniform [0,1/2].Does this have to be continuous?
John's solution is correct. The solution I had was not continuous as I originally thought.
I guess this turned out to be a lame question =)
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