Uniform pdf from difference of two stochastic variables?

[bemortu]

Hi,

I'm trying to find a probability distribution (D) with the following property:
Given two independent stochastic variables X₁ and X₂ from the distribution D, I want the difference Y=X₁-X₂ to have a uniform distribution (one the interval [0,1], say).

I don't seem to be able to solve it. I'm not even sure that such a distribution exists...

Any ideas?

 

[mathman]

Sugestion (outline). For simplicity I will make it uniform on the interval (-1/2,1/2). The characteristic function is sin(t/2)/(t/2). Take the square root and then the inverse Fourier transform should give you something close to what you want (the sum of two random variables will have a uniform distribution).

 

[bemortu]

Yes, that's one of the things I already tried. The problem is that I didn't manage to calculate that inverse Fourier transform. I tried it with Mathematica, which could not find an analytical solution. I also tried the numerical inverse Fourier transform in Mathematica but it also failed. Maybe it means that this distribution doesn't exist?

 

[mathman]

My guess: there is no density function. You might try getting the distribution function.

F(y) - F(x) = 1/2π ∫{(exp(ity) - exp(itx))φ(t)/(it)}dt
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