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The author is not solving the same problem he states in the title. I'm not even sure what problem he is solving. For one thing, the pdf of uniform distribution

by xyzzyz 14y ago

The author is not solving the same problem he states in the title. I'm not even sure what problem he is solving. For one thing, the pdf of uniform distribution on a [0, 1] with regard to Lebesgue measure on R is a function that's 1 on [0, 1] and 0 elsewhere. The pdf he gave is actually a pdf of Poisson distribution with regard to counting measure. Using simple methods from martingale theory we can show that the expected value we're looking for must be no smaller than 2 and no larger than 4, but I'm in no mood to delve into more detailed calculations.

abraxasz 14y ago
Yeah, I tried to play work out the solution before looking at the article, but realized that the problem was not even well posed. You could ask for the number of uniforms needed so that the probability of their convolution being greater than one exceeds a certain value, but that's not what he is asking. And yes, he doesn't work with uniform distributions, but Poissons. Weird article.
drostie 14y ago
You could probably pull out the exact probability mass function with characteristic functions. To review why these would be useful, let χ[T](s) = ⟨ exp(i ω T) ⟩, then χ[A + B] = χ[A] χ[B] for any A and B which are independent, so the characteristic function for a uniform random variable is going to be χ = (exp(i ω) − 1)/(i ω) and thus the sum of n independent uniform random numbers is just χⁿ. Meanwhile the PDF of a random variable is just the inverse Fourier transform of the characteristic function, so we have fₙ(t) = ∫ dω/(2π) exp(-i ω t) [χ(ω)]ⁿ Pr(n samples > 1) = ∫{1 → ∞} dt fₙ(t). You could then interchange those two integrals, solving the convergence problem of exp(-i ω ∞) by giving ω an infinitesimal negative-imaginary component. This translates to moving the ω contour a little below the real line, and then the normal tricks of complex analysis should give you definite values for all of the integrals as residues about the simple pole at 0.
mturmon 14y ago
Agree. The relevant concepts from the theory of stochastic processes would be "stopping time" or "expected hitting time". There are lots of relationships between the jump times of the Poisson process and subdivisions of the real line, so it's possible that the post is doing some version of a valid calculation, but without the right reasoning/understanding.

by xyzzyz 14y ago

abraxasz 14y ago

Yeah, I tried to play work out the solution before looking at the article, but realized that the problem was not even well posed. You could ask for the number of uniforms needed so that the probability of their convolution being greater than one exceeds a certain value, but that's not what he is asking. And yes, he doesn't work with uniform distributions, but Poissons. Weird article.

drostie 14y ago

You could probably pull out the exact probability mass function with characteristic functions. To review why these would be useful, let χ[T](s) = ⟨ exp(i ω T) ⟩, then χ[A + B] = χ[A] χ[B] for any A and B which are independent, so the characteristic function for a uniform random variable is going to be χ = (exp(i ω) − 1)/(i ω) and thus the sum of n independent uniform random numbers is just χⁿ. Meanwhile the PDF of a random variable is just the inverse Fourier transform of the characteristic function, so we have fₙ(t) = ∫ dω/(2π) exp(-i ω t) [χ(ω)]ⁿ Pr(n samples > 1) = ∫{1 → ∞} dt fₙ(t). You could then interchange those two integrals, solving the convergence problem of exp(-i ω ∞) by giving ω an infinitesimal negative-imaginary component. This translates to moving the ω contour a little below the real line, and then the normal tricks of complex analysis should give you definite values for all of the integrals as residues about the simple pole at 0.

mturmon 14y ago

Agree. The relevant concepts from the theory of stochastic processes would be "stopping time" or "expected hitting time". There are lots of relationships between the jump times of the Poisson process and subdivisions of the real line, so it's possible that the post is doing some version of a valid calculation, but without the right reasoning/understanding.