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Generating Biased Random Numbers

Problem

Instead of a truly random number, you wish to randomly select a value from a set in which some values are more likely than others. For example, you may wish to simulate a normal distribution (i.e., a "bell curve") for a set of data.

Solution

We will give a recipe for generating numbers with a normal distribution (aka Gaussian distribution, the bell shaped one). A library is under way see: Schemathics CVS, but we will discuss a do-it-yourself method for explanatory purposes.

You will have to determine what kind of distribution you want, and locate the appropriate algorithm from a statistics reference.

For this recipe, we will consider the normal (Gaussian) distribution. If you need other distributions see either the CVS or consult a numerical analyst.

The function make-normal-distributed-variable returns a stochastic variable (a thunk) with mean mu and standard deviance sigma.:

; 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

> (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 (present in the code as (random-real)) and generates two numbers with a mean of my and standard deviation sigma. Note that the method produces two numbers at a time. 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 Schemathics, 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.

Note

This recipe needs updating. Nowadays it is better to use either

The PLT Scheme Scientific Collection

or the random number package from Schematics

random.plt from Schematics

Hmm. It seems someone out to resubmit random.plt to PlaneT? 3xx.

-- JensAxelSoegaard - 11 Dec 2006

Contributors

-- BrentAFulgham - 14 May 2004

-- JensAxelSoegaard - 01 Jun 2004

[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: