## Monday, April 07, 2008

### Another Random Math Puzzle

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.

Keenan said...

Easy: X is defined by a uniform distribution over [0,1]; Y is defined by a Dirac delta distribution at 0.

Keenan said...

...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?

Yisong said...

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.

Keenan said...

Ok, so I was tired and forgot about dice. :)

John Carrino said...

How about a coinflip choice of 0 or .5 and a uniform [0,1/2].

Does this have to be continuous?

Yisong said...

John's solution is correct. The solution I had was not continuous as I originally thought.

Yisong said...

I guess this turned out to be a lame question =)