Let U be a uniform random variable taking values between 0 and 1 inclusive. Find two independent random variables X and Y such that U = X + Y, or prove that it is impossible.

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.

## 7 comments:

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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