Instead of producing a series of numbers distributed uniformly over an interval, we
need data following one of the classical distributions such as the normal distribution
(i.e. the numbers should give a "bell curve").
The easy solution is use the
"random.plt" library from
PLaneT.
Let's try generating numbers with a Gaussian distribution (a normal distribution
with mean 0 and standard deviance 1):
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If we want to mimick a stocastic variable, we can use
random-source-make-gaussians and
a source of random bits.
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Alternatively, one could simple define
X as
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Given a distribution, lookup an algorithm in a statistics reference.
If you can't find an algorithm, consult a numerical analyst.
Let's examine the case of the normal distribution. The two parameters
mu (mean)
and
sigma (standard deviance) determines a specific normal distribution.
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An example of usage:
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If you are unsatisfied with the fact that you get the exact same numbers as above,
then randomize the source of random numbers:
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The algorithm used is the polar Box Muller method. The algorithm takes two independent uniformly distributed random numbers between 0 and 1 (represented in the code as
(random-real)) and generates two numbers with a mean of my and standard deviation sigma. The method produces two numbers at a time, so since we only need one, the second is saved for later in the variable
next.
Note that the Perl Cookbook includes an interesting discussion of converting a set of values (and weights) into a distribution. This should also be converted to Scheme and shown here.
Mathematically-inclined Schemers should also take a good look at
random.ss, which
contains these and many other statistical methods.
It's also worth noting that if a bell-curve type thing is all you're looking for, generating two or more random numbers and taking the average will tend to favor the middle values. For example, consider a pair of dice: there is exactly one combination out of 36 that yields 2 and one that yields 12 (the outlying values), while there are six combinations that yield 7 (the center value). You could also use a weighed average to reduce the effect if averaging two random numbers produces a bell curve which is too steep for your application.
--
BrentAFulgham - 14 May 2004
--
JensAxelSoegaard - 01 Jun 2004
--
JensAxelSoegaard - 12 Dec 2006
[TODO: Move the following remarks to another recipe]
If you wish to randomly select from a set of weights and values, convert the weights into a probability distribution, then use the resulting distribution to pick a value.
If you have a list of weights and values you want to randomly pick from, follow this two-step process: First, turn the weights into a probability distribution with weight_to_dist below, and then use the distribution to randomly pick a value with weighted_rand:
[TODO: Use the random-source-make-discretes from random.ss to solve the above problem]
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