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Generating Biased Random Numbers

Problem

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").

Solution

The library solution

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):

> (require (planet "random.ss" ("schematics" "random.plt" 1 0)))
> (random-gaussian)
0.7386912134436788
> (random-gaussian)
-0.4388994504610697

If we want to mimick a stocastic variable, we can use random-source-make-gaussians and a source of random bits.

> (define X (random-source-make-gaussians default-random-source))
> (X)
0.5826066449247809
> (X)
0.7865269446783535

Alternatively, one could simple define X as

> (define X (lambda () (random-gaussian)))

The do it your self solution

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.

; derived from example in the documentation of SRFI27
(require (lib "27.ss" "srfi"))

(define (make-normal-distributed-variable mu sigma)
  (let ((mu    (* 1.0 mu))
        (sigma (* 1.0 sigma))
        (next  #f))
    (lambda ()
      (cond
        (next  (let ((result next))
                 (set! next #f)
                 (+ mu (* sigma result))))
        (else  (let loop ()
                 (let* ((v1 (- (* 2.0 (random-real)) 1.0))
                        (v2 (- (* 2.0 (random-real)) 1.0))
                        (s  (+ (* v1 v1) (* v2 v2))))
                   (cond
                     ((>= s 1.0) (loop))
                     (else       (let ((scale (sqrt (/ (* -2.0 (log s)) s))))
                                   (set! next (* scale v2))
                                   (+ mu (* sigma scale v1))))))))))))

An example of usage:

> (define X (make-normal-distributed-variable 0 1))
> (X)
0.7386912134436788
> (X)
-0.4388994504610697
> (X)
0.5826066449247809

If you are unsatisfied with the fact that you get the exact same numbers as above, then randomize the source of random numbers:

> (random-source-randomize! default-random-source)

Discussion

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.

Comments about this recipe

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.

Contributors

-- 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]