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The following archive was created by hippie-mail 7.98617-22 on Fri Aug 21 1998 - 17:20:14 ADT
Enzo Michelangeli (em@who.net)
Fri, 10 Jul 1998 12:26:39 +0800
-----Original Message-----
From: Mok-Kong Shen <mok-kong.shen@stud.uni-muenchen.de>
To: CodherPlunks@toad.com <CodherPlunks@toad.com>
Date: Wednesday, July 08, 1998 5:26 PM
Subject: Re: Random Data from Geiger Counter
>Perry E. Metzger wrote:
>
>> My big question is this: are there tools for taking a set of random
>> numbers dispersed according to a non-uniform distribution, like a
>> poisson or normal distribution, and turning them into a set of random
>> numbers over a uniform distribution? Given such tools, timing
>> intervals between the geiger counter ticks is probably safe --
>> otherwise, it may skew the results subtly.
>
>If you have a (really) uniform distribution in [0,1) and add to
>it another arbitrary distribution in [0,1) and take mod 1, the result
>is a uniform distribution. This sounds strange but is true.
>
>M. K. Shen
Yes, and this can be extended by induction to N distributions: unlike what
it happens with the simple sum of random variables, whose distribution comes
to approximate a gaussian bell curve (Central Limit Theorem), the modular
addition of uniformly-distributed variables maintains a uniform
distribution. The simple sum has always a distribution which is the
convolution of the two individual distributions: in our case, a triangle
rising from (x=0, y=0) to (x=1, y=1) and then decreasing to (x=2, y=0). If
you take the mod 1, the left side of the triangle is added to the right side
(because e.g. both 1.3 and 0.3 in the sum contribute to 0.3 in the sum
modulo 1), and we again have a rectangular shape.
By the way, this also supplies a proof that low bits (or digits) of a random
variable represent a new variable with a distribution very close to uniform,
regardless of the original distribution. For example, let's suppose that the
values of the original variable range between 0 and 10, with an oddly-shaped
distribution that obviously falls to zero outside that interval. Its
fractional part will have a distribution obtained by adding the heights of
the 10 trapezoids with x ranging from 0 to 1, from 1 to 2 etc. The "roof" of
each trapezoid can be approximated with a straight line. Some of those lines
will be rising, some will be falling; it turns out that the slants cancel
each
other in the sum, because adding them together is equivalent to integrate
the differentials of the original distribution, and that integral must be
zero because the distribution is zero at both extremes.
OK, this might be made more rigorous (and pictures would help), but you got
the idea.
Enzo
The following archive was created by hippie-mail 7.98617-22 on Fri Aug 21 1998 - 17:20:14 ADT