[math-fun] an almost-uniform random variable?
Here's a fun fact Oded Schramm just told me: If X, Y, Z are independent and uniform on [0,1], then so is (XY)^Z. And here's a fun fact related to Archimedes' insight about the sphere and the cylinder: If X, Y, Z are independent Gaussians of mean 0 and variance 1, X/(X^2+Y^2+Z^2) is uniform on [0,1]. Since I'm teaching a course on simulation of stochastic processes this coming semester, these could serve as good homework problems of the "do a simulation, make a conjecture, try to prove it" variety. But I'd like a third homework problem where some simple combination of random variables LOOKS like it's uniform but ISN'T, so the students don't come away with the impression that "if it's roughly flat, it must be uniform". So: Does anyone know of a probability distribution that's close to uniform (or close to Gaussian, or something like that), but isn't? Jim Propp
Date: Fri, 29 Aug 2008 17:46:58 -0400 From: James Propp <jpropp@cs.uml.edu>
Since I'm teaching a course on simulation of stochastic processes this coming semester, these could serve as good homework problems of the "do a simulation, make a conjecture, try to prove it" variety.
But I'd like a third homework problem where some simple combination of random variables LOOKS like it's uniform but ISN'T, so the students don't come away with the impression that "if it's roughly flat, it must be uniform".
So: Does anyone know of a probability distribution that's close to uniform (or close to Gaussian, or something like that), but isn't?
My 2 favorite problems of that sort are: (1) Figure out how to generate random numbers uniformly distributed on the (N-1)-sphere. Muller & Marsaglia's method for generating randoms uniformly distributed on the n-sphere shows that an N-vector of random standard normal variables, normalized to 1, is uniformly distributed on the (N-1)-sphere. Muller, M. E. "A Note on a Method for Generating Points Uniformly on N-Dimensional Spheres", Comm ACM 2 (1959), 19-20. Marsaglia, G. "Choosing a Point from the Surface of a Sphere", Ann Math Stat 43 (1972), 645-646. Marsaglia's email is: George Marsaglia <geo@stat.fsu.edu>. (The hypercube rejection method is tempting, but as N becomes large, the hypercube is "all corners" and you reject almost all trials! Rejection prob -> 1 as N -> +oo. So a related problem would be to get them to figure out WHY the rejection method is going wrong in high dimensions, taking forever to generate anything.) (2) Another "zinger" problem is to given them randoms from a Cauchy distribution ~ 1/(1+x^2), which looks more or less normal; then let them find out that the uncertainty on the mean goes UP with the number of samples. (Since the standard deviation is infinity, the central limit theorem doesn't apply.) Moral: you can always compute the mean of a finite sample; that usually gives you the mean of the density from which they're drawn -- except when it doesn't! J Romano & A Siegel, _Counterexamples in Probability and Statistics_, example 5.43, p. 115. http://books.google.com/books?id=irKSXZ7kKFgC&pg=PA115&lpg=PA115&dq=cauchy+a... -- Steve Rowley <sgr@alum.mit.edu> http://alum.mit.edu/www/sgr/ Skype: sgr000
So: Does anyone know of a probability distribution that's close to uniform (or close to Gaussian, or something like that), but isn't?
Dear Jim, assuming your students don't know about the central limit theorem, if all X_i are uniform on [0,1], then X_1 + X_2 + X_3 + ... + X_12 seems like the gaussian normal distribution N(6,1) but for rather trivial reasons isn't since there are no negative values. Best wishes, Stefan
participants (3)
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James Propp -
Stefan Steinerberger -
Steve Rowley