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.

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)

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`

.

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.

-- 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:

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TopicType: | Recipe |

ParentTopic: | NumberRecipes |

TopicOrder: | 100 |

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